Bessel and Related Functions: Volume 2 Numerical Results 9783110682472, 9783110681635

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Alexander Apelblat Bessel and Related Functions

Also of interest Bessel and Related Functions. Mathematical Operations with Respect to the Order Volume : Theoretical Aspects Alexander Apelblat,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ----, Set-ISBN ---- Combinatorial Functional Equations. Basic Theory Yanpei Liu,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ---- Combinatorial Functional Equations. Advanced Theory Yanpei Liu,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ---- Tensor Numerical Methods in Scientific Computing Series: Radon Series on Computational and Applied Mathematics  Boris N. Khoromskij,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ---- Fractional-Order Equations and Inclusions Series: Radon Series on Computational and Applied Mathematics  Michal Fečkan, JinRong Wang, Michal Pospíšil,  ISBN ----, e-ISBN (PDF) ----, e-ISBN (EPUB) ----

Alexander Apelblat

Bessel and Related Functions Mathematical Operations with Respect to the Order Volume 2: Numerical Results

Author Prof. Dr. Alexander Apelblat Ben-Gurion University of the Negev Department of Chemical Engineering Beer Sheva, Israel [email protected]

ISBN 978-3-11-068163-5 e-ISBN (PDF) 978-3-11-068247-2 e-ISBN (EPUB) 978-3-11-068347-9 Library of Congress Control Number: 2020933852 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2020 Walter de Gruyter GmbH, Berlin/Boston Cover image: Anon. (2010). Tinted detail of Friedrich Wilhelm Bessel [digital image]. http://www.dspace.cam.ac.uk/handle/1810/224200. University of Cambridge, Institute of Astronomy Library. CC BY 4.0. Licence: https://creativecommons.org/licenses/by/4.0/ Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

To Ira and Yoram

Contents 1

Numerical Results – Tabulation of First, Second and Third Derivatives with Respect to the Order of the Bessel and Modified Bessel Functions 1 1.1 First, Second and Third Derivatives with Respect to the Order of the Bessel Function of the First Kind 2 1.2 First, Second and Third Derivatives with Respect to the Order of the Bessel Function of the Second Kind 35 1.3 First, Second and Third Derivatives with Respect to the Order of the Modified Bessel Function of the First Kind 74 1.4 First, Second and Third Derivatives with Respect to the Order of the Modified Bessel Function of the Second Kind 109

2

Numerical Results – Tabulation of First, Second and Third Derivatives with Respect to the Order of the Struve, Modified Struve, Anger and Weber Functions 141 2.1 First, Second and Third Derivatives with Respect to the Order of the Struve Functions 141 2.2 First, Second and Third Derivatives with Respect to the Order of the modified Struve Functions 175 2.3 First, Second and Third Derivatives with Respect to the Order of the Anger Functions 210 2.4 First, Second and Third Derivatives with Respect to the Order of the Weber Functions 245

3

Numerical Results – Tabulation of the Kelvin Functions, Their First and Second Derivatives with Respect to the Order 281 3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives with Respect to the Order 281 3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives with Respect to the Order 315 3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives with Respect to the Order 349 3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives with Respect to the Order 389

4

Numerical Results – Tabulation of Functions, First and Second Derivatives with Respect to the Order of the Integral Bessel Functions 429 4.1 The Integral Bessel Functions of the First Kind and Their First and Second Derivatives with Respect to the Order 429

VIII

Contents

4.2 4.3 4.4

Index

Integral Bessel Functions of the Second Kind and Their First and Second Derivatives with Respect to the Order 463 The Integral Modified Bessel Functions of the Second Kind and Their First and Second Derivatives with Respect to the Order 502 The Bessel and Related Functions with the Same Argument and Order 541 551

1 Numerical Results – Tabulation of First, Second and Third Derivatives with Respect to the Order of the Bessel and Modified Bessel Functions The earliest short table of the first and second derivatives with respect to the order of the Bessel function Jν(t) was published in 1935 by Airey [42, 43, for list of References see Part 1]. He calculated ∂Jν(t)/∂ν and ∂2Jν(t)/∂ν2 with the argument of the Bessel function at steps t = 1, 2, 3, . . . , 20, but with the order ν in short ranges, t – 1 ≤ ν ≤ t + 1. The table is constructed with 0.1 intervals and four places of decimals. Airey also mentioned, in references, the existence of his tabulation from 1928, where values of the first derivatives are reported with respect to the order for ν equals ± 1/2 and 3/2. Lee and Radosevich [55] in 1960 calculated to four places the first derivatives with respect to the order for ν equals k + 1/4, k + 1/3, k + 2/3 and k + 3/4, k = 0, 1, 2, 3, 4 and 1 ≤ t ≤ 15 in 0.5 intervals. With the same precision, Erber and Gordon [49] in 1963 reported the first derivatives of the modified Bessel function of the first kind ∂Iν(t)/∂ν for ± 1/3 with 0 ≤ t ≤ 1 in 0.01 intervals, and with 1 ≤ t ≤ 5 in 0.05 intervals. From the literature search, it is evident that only a small number of tabulations of derivatives with respect to the order of the Bessel functions exist and these tables have rather accidental, sporadic character. In the case of the integral Bessel functions, even functions themselves are tabulated in a very limited form. In the Abramowitz–Stegun handbook [9], only the Bessel functions with zero order are tabulated. Their integrands are (1 – J0(t))/t, (1 – I0(t))/t, Y0(t)/t and K0(t)/t, and the integration limits vary in 0 ≤ t ≤ 5 region in 0.1 intervals. The Kelvin functions berν(t), beiν(t), kerν(t) and keiν(t) and their derivatives with respect to the argument t were tabulated by McLachlan [22] for integer orders n = 1–5 and arguments 0–10, mainly for the zero order. Considering this unfortunate situation in the literature, I decided to prepare in systematic way a set of tables, where the first, the second and the third derivatives are given with respect to the order of the Bessel functions, the modified Bessel functions, the Struve functions, the modified Struve functions and the Anger and Weber functions (Chapter 2). In the case of the integral Bessel functions (Chapter 4) and the Kelvin functions (Chapter 3), these functions, their first and second derivatives with respect to the order are tabulated. The orders cover 0 ≤ ν ≤ 5 range, with 0.05 intervals. The arguments of functions lie in 0.05 ≤ t ≤ 5 range with 0.05 intervals for t ≤ 1.0, and with 1.0 intervals for t > 1.0. These ranges of ν and t give the most characteristic patterns of computed derivatives and functions as can be observed from graphical representations presented in this book. Values of derivatives in tables are presented in the form to preserve six significant figures coming from computations. Numerical calculations were performed by using MATHEMATICA program. https://doi.org/10.1515/9783110682472-001

1 Numerical Results – Tabulation of First, Second and Third Derivatives

2

Derivatives with respect to the order were mainly calculated by determination of integrals in the integral representations of Bessel functions. In other cases, calculations were performed directly, by applying the central-difference formulas of order O(h4) with h = 0.001, with the Bessel and related functions taken from MATHEMATICA. The consistency of both methods of calculations was checked and was found to be very satisfactory. The reliability of calculations was also confirmed by comparison of results presented in Tables 1.1–1.6, reported by Airey [42, 43], Lee and Radosevich [55] and Erber and Gordon [49].

1.1 First, Second and Third Derivatives with Respect to the Order of the Bessel Function of the First Kind Changes in values of the first, second and third derivatives with respect to the order of the Bessel function of the first kind are limited, and occur only when ν and t are small. As can be observed in the following figures, the functional form of these derivatives is rather simple. Using MATHEMATICA program, the derivatives with respect to the order of the Bessel function of the first kind were determined by applying the following integral representations: ∂ Jν ðzÞ 1 = ∂ν π

ðπ t sinðz sin t −ν tÞdt 0

1 + π

e

t2 cosðz sin t −ν tÞ dt

1 + π

1 + π

½t sinðπ νÞ − π cosðπ νÞ dt

ðπ

∂ Jν ðzÞ 1 =− 2 ∂ν π

∂ Jν ðzÞ 1 =− ∂ν3 π

−z sinh t − νt

0

2

3

(1:1:1)

∞ ð

0 ∞ ð

(1:1:2) e

−z sinh t − νt

½ðπ − t Þ sinðπ νÞ + 2 πt cosðπ νÞ dt 2

2

0

ðπ t3 sinðz sin t −ν tÞ dt 0 ∞ ð

e

(1:1:3) −z sinh t − νt

½ðt − 3π tÞ sinðπ νÞ + ðπ − 3π t Þ cosðπ νÞ dt 3

2

3

2

0

Similar to the first derivatives, the second derivatives with respect to the order of the Bessel functions of the first kind ∂2Jν(t)/∂ν2 vary for small values of the order and argument, and they tend to be zero for larger values of ν and t (Figures 1.1 and 1.2).

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

3

0.0 –0.5

∂Jv(t) ∂v

–1.0 –1.5 –2.0 0.0 0.2 0.4 0.6

t

0.8

1.0

1.0 0.0

2.0

3.0

4.0

Figure 1.1: First derivatives with respect to the order of the Bessel functions of the first kind ∂Jν(t)/∂ν as a function of ν and t.

3.0

1 2.0

2 ∂2Jv(t) ∂v2

1.0 0.0

3 –1.0

5

8

7

6

4 –2.0 0.0

1.0

2.0

3.0

4.0

Figure 1.2: Second derivatives of the Bessel function of the first kind with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 0.25; 3 – t = 0.50; 4 – t = 1.0; 5 – t = 2.0; 6 – t = 3.0; 7 – t = 4.0; 8 – t = 5.0.

Third derivatives with respect to the order of the Bessel functions of the first kind ∂3Jν(t)/∂ν3, presented in Figures 1.3 and 1.4, show a more oscillatory character than lower derivatives, but essentially their behaviour is similar. It is worthwhile to note

1 Numerical Results – Tabulation of First, Second and Third Derivatives

4

that with increasing value of the order ν, for small arguments t (Figure 1.4) they have nearly zero value for ν > 1 (Figure 1.5).

3.0 2.0 1.0

∂2Jv(t) ∂v2

0.0 –1.0 –2.0 0.0 0.2 0.4

t

0.6 0.8 1.0

1.0

3.0

2.0

4.0

0.0

Figure 1.3: Second derivatives with respect to the order of the Bessel functions of the first kind ∂2Jν(t)/∂ν2 as a function of ν and t.

8.0

5 0.0

–8.0

∂3Jv(t) ∂v3

4 3

–16.0 –24.0

2

–32.0

1 –40.0 0.0

0.5

1.0

1.5

Figure 1.4: Third derivatives of the Bessel function of the first kind with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 0.10; 3 – t = 0.15; 4 – t = 0.20; 5 – t = 0.25.

5

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

4 2 0 ∂3Jv(t) ∂v3

–2 –4 –6 –8 –10 0.0 0.2 0.4

t

0.6 0.8 1.0 0.0

1.0

3.0

2.0

4.0

5.0

Figure 1.5: Third derivatives with respect to the order of the Bessel functions of the first kind ∂3Jν(t)/∂ν 3 as a function of ν and t.

Table 1.1: First derivatives with respect to the order of the Bessel functions of the first kind ∂Jν(t)/∂ν. t ν . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

6

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

7

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

8

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.1 (continued ) t ν

.

.

.

. . .

.

.

.

−.

−. −. −.

−. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

9

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

10

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

11

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

12

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

13

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.1 (continued ) t ν . .

.

.

.

.

.

.

−. −.

−. −.

−. −.

−. −.

−. −.

−. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

14

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

15

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

Table 1.2: Second derivatives with respect to the order of the Bessel functions of the first kind ∂2Jν(t)/∂ν2. t ν . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

−. . . . . . . . . . . . .

16

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.2 (continued ) t ν

.

.

.

. .

.

.

.

. .

. .

. .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

21

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t ν . .

.

.

.

.

.

.

−. −.

−. −.

−. −.

. .

. .

. .

23

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −.

24

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

25

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.2 (continued ) t ν . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. . .

Table 1.3: Third derivatives with respect to the order of the Bessel functions of the first kind ∂3Jν(t)/∂ν3. t ν . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

26

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. . . . . . . . . . . . . . −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . −. −. −.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . −. −. −.

27

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −.

t ν . .

.

.

.

.

.

.

. .

. .

. .

. .

. .

. .

28

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

29

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

30

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.3 (continued ) t ν . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . −.

. . . . . . . . . . . . −.

. . . . . . . . . . . . −.

. . . . . . . . . . . . −.

. . . . . . . . . . . . −.

. . . . . . . . . . . . −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

31

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −.

32

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −.

.

t ν . . . . . . . . .

.

.

.

.

.

. . . . . . . . .

. . . . . . . . .

−. −. −. −. −. . . . .

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

. . . . . . . . .

33

1.1 Derivatives with Respect to the Order of the Bessel Function of the First Kind

Table 1.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

34

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.3 (continued ) t ν . . . . . . . .

.

.

.

.

.

. . . . . . . −.

. . . . . . . −.

. . . . . . −. −.

. . . . . −. −. −.

. . . . . . . −.

. . . . . . . . .

1.2 First, Second and Third Derivatives with Respect to the Order of the Bessel Function of the Second Kind First three derivatives with respect to the order of the Bessel function of the second kind were determined directly using the central-difference formulas of order O(h4) with h = 10−3. The Bessel functions of the second kind Yν(t) were taken from MATHEMATICA program: ∂ Yν ðtÞ − Yν + 2 h ðtÞ + 8Yν + h ðtÞ − 8Yν − h ðtÞ + Yν − 2 h ðtÞ = ∂ν 12 h

(1:2:1)

∂ 2 Yν ðtÞ − Yν + 2 h ðtÞ + 16Yν + h ðtÞ − 30 Yν ðtÞ + 16 Yν − h ðtÞ − Yν − 2 h ðtÞ = ∂ ν2 12 h2

(1:2:2)

∂ 3 Yν ðtÞ − Yν + 3 h ðtÞ + 8Yν + 2 h ðtÞ − 13Yν + h ðtÞ 13Yν − h ðtÞ − 8Yν − 2 h ðtÞ + Yν − 3 h ðtÞ = + ∂ν3 8h3 8h3 (1:2:3) As shown in Figure 1.6, derivatives of the Bessel function of the second kind with respect to the order have a very large negative values for large ν and t.

1 Numerical Results – Tabulation of First, Second and Third Derivatives

36

0 –1,000

∂Yv(t) ∂v

–2,000 –3,000

0.0 0.2 0.4 0.6 t

0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 1.6: First derivatives with respect to the order of the Bessel functions of the second kind ∂Yν(t)/∂ν as a function of ν and t.

Behaviour of second and third derivatives with respect to the order of the Bessel functions of the first kind is very similar to that of the first derivatives (Figures 1.7 and 1.8).

0 –400 –800

𝜕 2Y

𝜈(t)

𝜕𝜈2

–1,200 –1,600

0.0 0.2 0.4 0.6 t

0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 1.7: Second derivatives with respect to the order of the Bessel functions of the second kind ∂2Yν(t)/∂ν2 as a function of ν and t.

37

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

0

–200

𝜕 3Y𝜈(t) 𝜕𝜈3

–400

–600

0.0 0.2 0.4 t

0.6 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 1.8: Third derivatives with respect to the order of the Bessel functions of the second kind ∂3Yν(t)/∂ν3 as a function of ν and t.

Table 1.4: First derivatives with respect to the order of the Bessel functions of the second kind ∂Yν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

38

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −,

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,.

39

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−, −, −, −, −, −.E+ −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −.

−,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −. −.E+

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −. −. −. −.E+ −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −. −. −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −,

t ν . .

.

.

.

.

.

−. −.

−. −.

−. −.

−. −.

−. −.

40

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

41

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

42

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.4 (continued ) t ν . . . . . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,. −, −, −, −, −, −, −, −,

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −,

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

43

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

44

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,.

t ν . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

45

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

46

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

47

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.4 (continued ) t ν . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,.

−. −. −,. −,. −,.

−. −. −. −. −.

−. −. −. −. −.

−. −. −. −. −.

.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

49

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

Table 1.5: Second derivatives with respect to the order of the Bessel functions of the second kind ∂2Yν(t)/∂ν2. t ν . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −.

.

.

. . . −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −.

. . . . . . −. −. −. −. −. −.

. . . . . . . −. −. −. −. −.

50

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −,

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,.

51

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −, −, −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −.

.

.

.

.

−,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −. −. −.E+ −. −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −. −. −. −.E+ −. −. −. −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −, −. −.E+ −.

52

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.5 (continued ) t ν . . . .

. −. −.E+ −. −.

.

.

−. −. −. −.

−. −. −. −.

. −. −. −. −.E+

. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

53

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,.

54

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −, −.E+ −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −,

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −,

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −,

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

t ν . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . −. −. −. −. −. −.

. . . . . . . . . . . . . . −. −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −.

. . . . . . . . . . . . . . . . . −. −.

55

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

56

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

.

.

.

t ν . .

.

.

. .

. .

. .

. .

. .

57

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

58

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

59

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,.

t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

60

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

61

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.6: Third derivatives with respect to the order of the Bessel functions of the second kind ∂3Yν(t)/∂ν3. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

63

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.E+ −. −.

.

.

.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −. −. −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −.

. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −,

64

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.6 (continued ) t ν . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −.E+

−. −. −.E+ −. −. −. −. −. −. −. −. −. −.E+ −. −. −.

−.E+ −. −. −. −. −. −. −. −. −. −. −. −.E+ −. −. −.

−. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −.

−, −, −.E+ −. −. −. −. −. −. −. −. −. −. −. −.E+ −.

t ν . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

65

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,.

66

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −, −.E+ −. −. −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −, −, −, −, −,

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −, −, −, −, −,

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −, −, −, −, −,

t ν . . . . .

.

.

.

.

.

−. −. −. −. −.

−. −. −. −. −.

. −. −. −. −.

. . −. −. −.

. . . −. −.

67

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

68

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

69

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.6 (continued ) t ν . . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,. −,. −,. −,. −, −,

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

70

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

71

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

.

.

.

−. −. −. . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

72

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

73

1.2 Derivatives with Respect to the Order of the Bessel Function of the Second Kind

Table 1.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

74

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.6 (continued ) t ν . . .

.

.

.

.

.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

1.3 First, Second and Third Derivatives with Respect to the Order of the Modified Bessel Function of the First Kind Derivatives with respect to the order of the modified Bessel function of the first kind were determined using MATHEMATICA program by applying the following integral representations: ðπ

∂ Iν ðzÞ 1 =− ∂ν π

t ez cos t sinðν tÞ dt + 0

1 π

∞ ð

e −z cosh t − νt ½t sinðπ νÞ − π cosðπ νÞ dt

0

(1:3:1) ∂ 2 Iν ðzÞ 1 =− ∂ν2 π

ðπ t2 ez cos t cosðν tÞ dt 0

1 + π ∂ 3 Iν ðzÞ 1 = ∂ν3 π

∞ ð

(1:3:2) e −z cosh t − ν t ½ðπ2 − t2 Þ sinðπ νÞ + 2 π t cosðπ νÞ dt

0

ðπ t3 ez cos t sinðν tÞ dt 0

1 − π

∞ ð

(1:3:3) e −z cosh t − ν t ½ðt3 − 3π2 tÞ sinðν tÞ + ðπ3 − 3 π t2 Þ cosðν tÞ dt

0

The form of derivatives with respect to the order of the modified Bessel function of the first kind differ from those of the first kind ∂Jν(t)/∂ν and higher derivatives (Tables 1.7–1.12). Figures 1.9–1.17 show that these derivatives tend quickly to zero for small values of the order ν and argument t.

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

75

0.0

1

5 4

–0.5

𝜕I𝜈(t)

3

𝜕𝜈 –1.0

2 –1.5

0.0

1.0

2.0

3.0

4.0

Figure 1.9: First derivatives of the modified Bessel function of the first kind with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 0.25; 3 – t = 0.50; 4 – t = 1.0; 5 – t = 1.50.

0.0 –0.5

𝜕I𝜈(t) 𝜕𝜈

–1.0 –1.5 –2.0 0.0 0.2 0.4 t 0.6 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 1.10: First derivatives with respect to the order of the modified Bessel functions of the first kind ∂Iν(t)/∂ν as a function of ν and t.

1 Numerical Results – Tabulation of First, Second and Third Derivatives

76

0.0

–2.0 𝜕I𝜈(t) 𝜕𝜈

–4.0 –6.0 0.0 1.0 2.0 3.0

t

4.0 5.0

0.0

2.0

1.0

4.0

3.0

5.0

Figure 1.11: First derivatives with respect to the order of the modified Bessel functions of the first kind ∂Iν(t)/∂ν as a function of ν and t.

8.0

1 6.0

𝜕2I𝜈(t) 𝜕𝜈2

4.0 2.0

2

0.0 –2.0 0.0

3 4 5 0.5

1.0

1.5

2.0

Figure 1.12: Second derivatives of the modified Bessel function of the first kind with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 0.25; 3 – t = 0.50; 4 – t = 1.0; 5 – t = 1.50.

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

4.0

2.0

𝜕2I𝜈(t) 𝜕𝜈2

0.0

–2.0

0.0

3.0 0.2

0.4

t

2.0 0.6

1.0

0.8 1.0 0.0

Figure 1.13: Second derivatives with respect to the order of the modified Bessel functions of the first kind ∂2Iν(t)/∂ν2 as a function of ν and t.

8.0

4.0 𝜕2I

𝜈(t)

𝜕𝜈2

0.0 –4.0 0.0 1.0 2.0 3.0 t 4.0 5.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 1.14: Second derivatives with respect to the order of the modified Bessel functions of the first kind ∂2Iν(t)/∂ν2 as a function of ν and t.

77

1 Numerical Results – Tabulation of First, Second and Third Derivatives

78

8.0

3 4 5 0.0

2

𝜕3I𝜈(t) 𝜕𝜈

3

–8.0

1 0.0

0.5

1.0

1.5

2.0

Figure 1.15: Third derivatives of the modified Bessel function of the first kind with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 0.25; 3 – t = 0.50; 4 – t = 1.0; 5 – t = 1.50.

5.0

𝜕3I𝜈(t) 𝜕𝜈3

0.0

–5.0 –10.0 0.0 0.2 0.4 0.6 t 0.8

1.0

1.0

2.0

3.0

0.0

Figure 1.16: Third derivatives with respect to the order of the modified Bessel functions of the first kind ∂3Iν(t)/∂ν3 as a function of ν and t.

79

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

0.0

𝜕3I𝜈(t) 𝜕𝜈3

–5.0

–10.0

0.0 1.0 2.0 3.0 t

4.0 5.0

1.0

0.0

4.0

3.0

2.0

Figure 1.17: Third derivatives with respect to the order of the modified Bessel functions of the first kind ∂3Iν(t)/∂ν3 as a function of ν and t.

Table 1.7: First derivatives with respect to the order of the modified Bessel functions of the first kind ∂Iν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

80

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

81

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

Table 1.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . .

.

.

.

.

.

.

−. −.

−. −.

−. −.

−. −.

−. −.

−. −.

82

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

83

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

Table 1.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

84

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.7 (continued ) t ν . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

85

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

Table 1.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

86

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

87

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

Table 1.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

88

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

89

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

Table 1.7 (continued ) t ν . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

Table 1.8: Second derivatives with respect to the order of the modified Bessel functions of the first kind ∂2Iν(t)/∂ν2. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

90

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

Table 1.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t ν . . . . . . . . .

.

.

.

.

.

.

−. . . . . . . . .

−. −. . . . . . . .

−. −. −. . . . . . .

−. −. −. −. . . . . .

−. −. −. −. −. . . . .

−. −. −. −. −. −. . . .

92

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

Table 1.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.8 (continued ) t ν . . . . . .

. . . . . . .

. . . . . . .

.

.

. . . . . .

.

.

. . . . . .

. . . . . .

. . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

95

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

Table 1.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

97

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

Table 1.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

98

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

Table 1.9: Third derivatives with respect to the order of the modified Bessel functions of the first kind ∂3Iν(t)/∂ν3. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

100

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

101

Table 1.9 (continued ) t ν . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −.

102

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

103

Table 1.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . .

.

.

.

.

.

.

. . .

. . .

. . .

. . .

. . .

. . .

104

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

105

Table 1.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

106

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.9 (continued ) t ν . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . −.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

1.3 Derivatives with Respect to the Order of the Modified Bessel Function: First Kind

107

Table 1.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

109

1.4 First, Second and Third Derivatives with Respect to the Order of the Modified Bessel Function of the Second Kind Using MATHEMATICA program, derivatives with respect to the order of the modified Bessel function of the second kind were determined by applying the following integral representations: ∂ Kν ðzÞ = ∂ν

∞ ð

∂ 2 Kν ðzÞ = ∂ν2 ∂ 3 Kν ðzÞ = ∂ν3

te −z cosh t sinhðνtÞ dt

0 ∞ ð

0 ∞ ð

(1:4:1)

t2 e −z cosh t coshðν tÞ dt

(1:4:2)

t3 e −z cosh t sinhðν tÞ dt

(1:4:3)

0

In calculated range of the order ν and the argument t, contrary to other derivatives with respect to the order of the Bessel functions, the derivatives of the modified Bessel functions of the second kind have complex form with distinct extremal values (Figures 1.18–1.20).

1 × 106 8 × 105 6 × 105

𝜕K𝜈(t) 𝜕𝜈

4 × 105 2 × 105 0.0 0.2 0.4 0.6 t

0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 1.18: First derivatives with respect to the order of the modified Bessel functions of the second kind ∂Kν(t)/∂ν as a function of ν and t.

110

1 Numerical Results – Tabulation of First, Second and Third Derivatives

1 × 106 8 × 105

𝜕2K𝜈(t) 𝜕𝜈2

6 × 105 4 × 105 2 × 105 0.0 0.2 0.4

t

0.6 0.8

2.0

1.0

1.0 0.0

4.0

3.0

5.0

Figure 1.19: Second derivatives with respect to the order of the modified Bessel functions of the second kind ∂2Kν(t)/∂ν2 as a function of ν and t.

1 × 106 8 × 105

𝜕3K𝜈(t) 𝜕𝜈3

6 × 105 4 × 105 2 × 105 0.0 0.2 0.4 0.6 t 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 1.20: Third derivatives with respect to the order of the modified Bessel functions of the second kind ∂3Kν(t)/∂ν3 as a function of ν and t.

111

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

Table 1.10: First derivatives with respect to the order of the modified Bessel functions of the second kind ∂Kν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. , , , , , , , , , , .E+ . . . . . . . . .E+ . . . . . . . . . .E+ . . . . . . . . .E+ . . . .

.

.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , .E+ . . . . . . . . . .E+ . . . . . . . . . . .E+

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , . E+ . . . . . . . . . . .E+ . .

. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , .E+ . . . . . .

.

.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , , .E+

. . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , ,

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

113

Table 1.10 (continued ) t ν . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . .E+ . . . . . . . .E+ . .

. . . . . . . . . .E+ . . . . .

. . . . . . . . .E+ . . . . . .

. . . . . .E+ . . . . . . . . .

. . . . . . . . . . . .E+ . . .

, , , , .E+ . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

114

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,.

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

115

Table 1.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , , .E+ . . . . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , , .E+ .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , ,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , ,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , ,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,

t ν . . . . . . . .

.

.

.

.

.

.

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

116

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

117

Table 1.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

118

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.10 (continued ) t ν . . . . .

.

.

.

.

.

.

,. ,. ,. ,. ,.

,. ,. ,. ,. ,.

,. ,. ,. ,. ,.

,. ,. ,. ,. ,.

,. ,. ,. ,. ,.

,. ,. ,. ,. ,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

119

Table 1.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . .

.

.

.

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

Table 1.11: Second derivatives with respect to the order of the modified Bessel functions of the second kind ∂2Kν(t)/∂ν2. t ν . . . . . . . . . .

. . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

121

Table 1.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , ,

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. , , , , , , , .E+ . . . . . . . . . .E+ . . . . . . . . .E+ . . . . . . . . .E+ . . . . . .

.

.

.

.

.

,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , .E+ . . . . . . . . . . .E+ . . . . . . . . . .E+ . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , .E+ . . . . . . . . . . .E+ . . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , .E+ . . . . . . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , .E+ .

. . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , ,

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

123

Table 1.11 (continued ) t ν . . . . . . .

. . . .E+ . . . .

. . . . . . . .

. . . . . . . .

. . . . . .E+ . .

. . . . . . . .

. , , , .E+ . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

125

Table 1.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , ,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , ,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , ,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

t ν . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

126

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

127

Table 1.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,.

128

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.11 (continued ) t ν . . . . .

.

.

.

.

.

.

,. ,. ,. ,. ,.

,. ,. ,. ,. ,.

,. ,. ,. ,. ,.

,. ,. ,. ,. ,.

,. ,. ,. ,. ,.

,. ,. ,. ,. ,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

129

Table 1.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

Table 1.12: Third derivatives with respect to the order of the modified Bessel functions of the second kind ∂3Kν(t)/∂ν3. t ν . . . . . . . .

. . . . . . . . .

.

.

.

.

.

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

131

Table 1.12 (continued ) t ν

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , ,

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,.

132

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

, , , .E+ . . . . . . . . .E+ . . . . . . . . . .E+ . . . . . . . . .E+ . ×  . . . . . . . .E+ . .

,. ,. ,. , , , , , , , , , , , .E+ . . . . . . . . . .E+ . . . . . . . . . . .E+ . . . . . . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , .E+ . . . . . . . . . . .E+ . . . . . . . . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , .E+ . . . . . . . . . . . .E+ .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , , .E+ . . . . . . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , , .E+ . .

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

133

Table 1.12 (continued ) t ν . . . . . . . .

. . . . . . .E+ . .

.

.

. . .E+ . . . . .

. .E+ . . . . . .

. . . . . . . . .

. . . . . .E+ . . .

. . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,.

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

135

Table 1.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , , .E+ . . . . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , , , , .E+ .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , , , , , ,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , , , , , ,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. , , , ,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,

t ν . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

136

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

137

Table 1.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

138

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.12 (continued ) t ν . . .

.

.

.

.

.

,. ,. ,.

,. ,. ,.

,. ,. ,.

,. ,. ,.

,. ,. ,.

. ,. ,. ,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4 Derivatives with Respect to the Order of the Modified Bessel Function

139

Table 1.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140

1 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 1.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . .

.

.

.

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

2 Numerical Results – Tabulation of First, Second and Third Derivatives with Respect to the Order of the Struve, Modified Struve, Anger and Weber Functions Derivatives with respect to the order of these Bessel-related functions are not tabulated in literature. The results presented here cover 0 ≤ ν ≤ 5 range, with 0.05 intervals. The arguments of functions are in 0.05 ≤ t ≤ 5 range, with 0.05 intervals for t ≤ 1.0 and with 1.0 intervals for t > 1.0. Calculations were performed using MATHEMATICA program. Derivatives with respect to the order were calculated by evaluation of integrals demonstrating the Anger and Weber integral representations. In other cases, they were performed directly by using the central-difference formulas of order O(h4) with h = 0.001, when the involved functions were taken from MATHEMATICA program.

2.1 First, Second and Third Derivatives with Respect to the Order of the Struve Functions Derivatives with respect to the order of the Struve functions were determined using central-difference formulas of order O(h4) with h = 10−3. The Struve functions Hν(t) were taken from MATHEMATICA program: ∂ Hν ðtÞ − Hν + 2 h ðtÞ + 8Hν + h ðtÞ − 8Hν − h ðtÞ + Hν − 2 h ðtÞ = ∂ν 12h

(2:1:1)

∂ 2 Hν ðtÞ − Hν + 2 h ðtÞ + 16Hν + h ðtÞ − 30Hν ðtÞ + 16Hν − h ðtÞ − Hν − 2 h ðtÞ = ∂ν2 12 h2

(2:1:2)

∂ 3 Hν ðtÞ − Hν + 3 h ðtÞ + 8Hν + 2 h ðtÞ − 13 Hν + h ðtÞ = ∂ ν3 8h3

(2:1:3)

13 Hν − h ðtÞ − 8Hν − 2 h ðtÞ + Hν − 3 h ðtÞ + 8h3

Functional behaviour of derivatives with respect to the order of the Struve functions resembles the Bessel functions of the first kind; the oscillatory character is damped and it tends to zero values with an increase in the order ν and the argument t (see Figures 2.1–2.18; Tables 2.1–2.12).

https://doi.org/10.1515/9783110682472-002

2 Numerical Results – Tabulation of First, Second and Third Derivatives

142

1.2 6 0.8 𝜕H󰜈(t)

4

0.4

𝜕󰜈

5

3

0.0 1

2

–0.4

0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.1: First derivatives of the Struve function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.50; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

0.1 0.0 –0.1

∂Hν(t)

–0.2

∂ν

–0.3 –0.4 –0.5

0.0

0.2

0.4

t

0.6

0.8

1.0

2.0

3.0

4.0

5.0

1.0 0.0

Figure 2.2: First derivatives with respect to the order of the Struve functions ∂Hν(t)/∂ν as a function of ν and t.

2.1 First, Second and Third Derivatives with Respect to the Order

143

1.0 0.8 0.6 ∂Hν(t) ∂ν

0.4 0.2 0.0 –0.2 –0.4

0.0 1.0 2.0 3.0 t 4.0 5.0 0.0

1.0

2.0

4.0

3.0

5.0

Figure 2.3: First derivatives with respect to the order of the Struve functions ∂Hν(t)/∂ν as a function of ν and t.

1.2 0.8 0.4

1 2

𝜕2H󰜈(t) 0.0 𝜕󰜈2

4

–0.4 –0.8

6

3

–1.2 0.0

5

1.0

2.0

3.0

4.0

5.0

Figure 2.4: Second derivatives of the Struve function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.50; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

2 Numerical Results – Tabulation of First, Second and Third Derivatives

144

0.6 0.4 0.2

𝜕2H

󰜈(t)

𝜕󰜈2

0.0 –0.2 –0.4 –0.6 –0.8

0.0 0.2 0.4 t 0.6 0.8 1.0

2.0

1.0

0.0

3.0

4.0

5.0

Figure 2.5: Second derivatives with respect to the order of the Struve functions ∂2Hν(t)/∂ν2 as a function of ν and t.

1.0 0.5

∂2H󰣍(t) 0.0 ∂󰣍2

–0.5 –1.0 0.0 1.0 2.0 t 3.0 4.0 5.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.6: Second derivatives with respect to the order of the Struve functions ∂2Hν(t)/∂ν2 as a function of ν and t.

2.1 First, Second and Third Derivatives with Respect to the Order

145

2.0

2 1.0

∂3H󰣍(t) ∂󰣍3

3 5

1 0.0

–1.0

4 6

–2.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.7: Third derivatives of the Struve function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.50; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

1.5 1.0 0.5

∂3Hv(t) ∂ν3

0.0 –0.5 –1.0

0.0 0.2 0.4 0.6 t 0.8 1.0 0.0

1.0

2.0

3.0

4.0

Figure 2.8: Third derivatives with respect to the order of the Struve functions ∂3Hν(t)/∂ν3 as a function of ν and t.

2 Numerical Results – Tabulation of First, Second and Third Derivatives

146

2.0 1.0

∂3H󰜈 (t) ∂󰜈3

0.0 –1.0 0.0 1.0

2.0

t

3.0

4.0 5.0 0.0

1.0

3.0

2.0

4.0

5.0

Figure 2.9: Third derivatives with respect to the order of the Struve functions ∂3Hν(t)/∂ν3 as a function of ν and t.

Table 2.1: First derivatives with respect to the order of the Struve functions ∂Hν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.1 First, Second and Third Derivatives with Respect to the Order

147

Table 2.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

148

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.1 (continued ) t ν

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −.

2.1 First, Second and Third Derivatives with Respect to the Order

149

Table 2.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

150

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.1 First, Second and Third Derivatives with Respect to the Order

151

Table 2.1 (continued ) t ν . . . . . . .

.

.

.

.

.

.

−. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

152

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.1 First, Second and Third Derivatives with Respect to the Order

153

Table 2.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

t ν . . . . . . . . . . . . .

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . −. −. −. −. −. −. −.

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

154

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

155

2.1 First, Second and Third Derivatives with Respect to the Order

Table 2.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

156

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.2: Second derivatives with respect to the order of the Struve functions ∂2Hν(t)/∂ν2. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 First, Second and Third Derivatives with Respect to the Order

157

Table 2.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

158

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.2 (continued ) t ν

.

.

.

. . . . . . . . . . . . . .

.

.

.

. . .

. . . . . . . . .

. . . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 First, Second and Third Derivatives with Respect to the Order

159

Table 2.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t ν . . . . . . . .

.

.

.

.

.

.

. . . . . . . .

. . . . . . . .

−. . . . . . . .

−. −. . . . . . .

−. −. −. . . . . .

−. −. −. −. . . . .

2.1 First, Second and Third Derivatives with Respect to the Order

161

Table 2.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

162

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1 First, Second and Third Derivatives with Respect to the Order

163

Table 2.2 (continued ) t ν . . . . . . .

.

.

. . . . . . .

. . . . . . .

.

.

. . . . . . .

.

.

. . . . . . .

. . . . . . .

. . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

164

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.1 First, Second and Third Derivatives with Respect to the Order

165

Table 2.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

Table 2.3: Third derivatives with respect to the order of the Struve functions ∂3Hν(t)/∂ν3. t ν . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

166

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.1 First, Second and Third Derivatives with Respect to the Order

167

Table 2.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

168

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.3 (continued ) t ν

.

.

.

. . . . .

.

.

.

−.

−. −. −. −. −.

−. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.1 First, Second and Third Derivatives with Respect to the Order

169

Table 2.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

170

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

t ν . . . . . . . . . . . . . .

.

.

.

. . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . −. −.

. . . . . . . . . . . . . −.

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

2.1 First, Second and Third Derivatives with Respect to the Order

171

Table 2.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

172

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.1 First, Second and Third Derivatives with Respect to the Order

173

Table 2.3 (continued ) t ν . . .

.

.

.

.

.

.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

.

.

.

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

2.2 First, Second and Third Derivatives with Respect to the Order

Table 2.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . .

2.2 First, Second and Third Derivatives with Respect to the Order of the modified Struve Functions The modified Struve functions Lν(t) were taken from MATHEMATICA program and their derivatives with respect to the order were evaluated by using the centraldifference formulas of order O(h4) with h = 10−3: ∂ Lν ðtÞ − Lν + 2 h ðtÞ + 8 Lν + h ðtÞ − 8Lν − h ðtÞ + Lν − 2 h ðtÞ = ∂ν 12 h

(2:2:1)

∂ 2 Lν ðtÞ − Lν + 2 h ðtÞ + 16Lν + h ðtÞ − 30Lν ðtÞ + 16 Lν − h ðtÞ − Lν − 2 h ðtÞ = ∂ν2 12 h2

(2:2:2)

∂ 3 Lν ðtÞ − Lν + 3 h ðtÞ + 8Lν + 2 h ðtÞ − 13 Lν + h ðtÞ = ∂ ν3 8h3 +

13 Lν − h ðtÞ − 8Lν − 2 h ðtÞ + Lν − 3 h ðtÞ 8h3

(2:2:3)

2 Numerical Results – Tabulation of First, Second and Third Derivatives

176

2.0

2

1

0.0

3

4

–2.0

∂L󰜈(t) ∂󰜈

5 6

–4.0 –6.0 –8.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.10: First derivatives of the modified Struve function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.50; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

0.6

0.4

∂L󰜈(t) ∂󰜈 0.2

0.0 0.0

0.2

0.4

t

0.6

0.8 1.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.11: First derivatives with respect to the order of the modified Struve functions ∂Lν(t)/∂ν as a function of ν and t.

2.2 First, Second and Third Derivatives with Respect to the Order

177

0.0

–2.0

∂L󰜈(t) ∂󰜈

–4.0 –6.0 0.0

1.0

2.0

t

3.0

4.0 5.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.12: First derivatives with respect to the order of the modified Struve functions ∂Lν(t)/∂ν as a function of ν and t.

4.0 2.0

3

1

0.0 ∂2L󰜈(t) ∂󰜈2

2

4

–2.0

5

–4.0 –6.0 –8.0 0.0

6

1.0

2.0

3.0

4.0

5.0

Figure 2.13: Second derivatives of the modified Struve function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.50; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

2 Numerical Results – Tabulation of First, Second and Third Derivatives

178

0.6 0.5 0.4

∂2L󰜈(t)

0.3

∂󰜈

0.2

2

0.1 0.0 0.0 0.2 0.4 t 0.6 0.8 1.0

1.0

0.0

3.0

2.0

4.0

Figure 2.14: Second derivatives with respect to the order of the modified Struve functions ∂2Lν(t)/∂ν2 as a function of ν and t.

2.0 0.0

∂2L󰜈(t) –2.0 ∂󰜈 2 –4.0 –6.0

0.0 1.0 t

2.0

3.0 4.0 5.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.15: Second derivatives with respect to the order of the modified Struve functions ∂2Lν(t)/ ∂ν2 as a function of ν and t.

2.2 First, Second and Third Derivatives with Respect to the Order

6

4.0

∂3L󰜈(t) ∂󰜈3

179

5

2.0 4 3

2

0.0 1

0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.16: Third derivatives of the modified Struve function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.50; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

1.5 1.0 0.5 ∂3L󰜈(t) ∂󰜈3

0.0 –0.5 –1.0

0.0 0.2 0.4 t 0.6 0.8 1.0 0.0

1.0

2.0

3.0

Figure 2.17: Third derivatives with respect to the order of the modified Struve functions ∂3Lν(t)/∂ν3 as a function of ν and t.

2 Numerical Results – Tabulation of First, Second and Third Derivatives

180

4.0 3.0 2.0

𝜕3L󰜈(t) 𝜕󰜈 3

1.0 0.0

–1.0 0.0 1.0 2.0 3.0 t 4.0 5.0 0.0

1.0

2.0

4.0

3.0

5.0

Figure 2.18: Third derivatives with respect to the order of the modified Struve functions ∂3Lν(t)/∂ν3 as a function of ν and t.

Table 2.4: First derivatives with respect to the order of the modified Struve functions ∂Lν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.2 First, Second and Third Derivatives with Respect to the Order

181

Table 2.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

182

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.4 (continued ) t ν

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

2.2 First, Second and Third Derivatives with Respect to the Order

183

Table 2.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

184

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.2 First, Second and Third Derivatives with Respect to the Order

185

Table 2.4 (continued ) t ν . . . . . .

.

.

.

.

.

.

−. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

186

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.2 First, Second and Third Derivatives with Respect to the Order

187

Table 2.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

188

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.2 First, Second and Third Derivatives with Respect to the Order

189

Table 2.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

190

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.5: Second derivatives with respect to the order of the modified Struve functions ∂ 2Lν(t)/∂ν2. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 First, Second and Third Derivatives with Respect to the Order

191

Table 2.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

192

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.5 (continued ) t ν

.

.

.

. . . . . . . . . . . . . . . . .

.

.

.

. . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

2.2 First, Second and Third Derivatives with Respect to the Order

193

Table 2.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

194

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t ν . . .

.

.

.

.

.

.

. . .

. . .

. . .

. . .

. . .

−. . .

2.2 First, Second and Third Derivatives with Respect to the Order

195

Table 2.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

196

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 First, Second and Third Derivatives with Respect to the Order

197

Table 2.5 (continued ) t ν . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

198

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 First, Second and Third Derivatives with Respect to the Order

199

Table 2.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

200

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.6: Third derivatives with respect to the order of the modified Struve functions ∂3Lν(t)/∂ν3. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.2 First, Second and Third Derivatives with Respect to the Order

201

Table 2.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

202

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.6 (continued ) t ν

.

.

. . . . . . . . . . . . . . . . .

.

.

.

.

−. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.2 First, Second and Third Derivatives with Respect to the Order

203

Table 2.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

204

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . .

.

.

.

.

.

.

. .

. .

. .

. .

. .

. .

2.2 First, Second and Third Derivatives with Respect to the Order

205

Table 2.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

206

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.2 First, Second and Third Derivatives with Respect to the Order

207

Table 2.6 (continued ) t ν . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . .

208

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 First, Second and Third Derivatives with Respect to the Order

209

Table 2.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2 Numerical Results – Tabulation of First, Second and Third Derivatives

210

2.3 First, Second and Third Derivatives with Respect to the Order of the Anger Functions Derivatives with respect to the order of the Anger functions were evaluated using MATHEMATICA program by applying the following integral representations: ∂ J ν ðzÞ 1 =− ∂ν π

ðπ

∂ 2 J ν ðzÞ 1 =− 2 ∂ν π ∂ J ν ðzÞ 1 = ∂ν3 π 3

t sinðνt − z sin tÞ dt

(2:3:1)

0

ðπ t2 cosðν t − z sin tÞ dt

(2:3:2)

0

ðπ t3 sinðνt − z sin tÞ dt

(2:3:3)

0

In the case of these functions, it is evident that in their oscillatory form, their values are smaller as ν and t increase and they tend to near zero values (Figures 2.19–2.36).

2.0

3 1.0

𝜕J󰜈(t) 𝜕󰜈

4

5

6

0.0

2

–1.0

1 –2.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.19: First derivatives of the Anger function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

2.3 First, Second and Third Derivatives with Respect to the Order

211

1.0 0.5 𝜕J󰜈(t) 0.0 𝜕󰜈 –0.5 –1.0

0.0 0.2 0.4 t 0.6 0.8 1.0

0.0

2.0

1.0

3.0

4.0

5.0

Figure 2.20: First derivatives with respect to the order of the Anger functions ∂Jν(t)/∂ν as a function of ν and t.

1.0 0.5

𝜕J󰜈(t) 𝜕󰜈

0.0 –0.5 –1.0

0.0 1.0 2.0 3.0 t 4.0 5.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.21: First derivatives with respect to the order of the Anger functions ∂Jν(t)/∂ν as a function of ν and t.

2 Numerical Results – Tabulation of First, Second and Third Derivatives

212

4.0

1

2.0

5

𝜕2J󰜈(t) 𝜕󰜈

2

6

0.0

4 –2.0

3 2

–4.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.22: Second derivatives of the Anger function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

2.0 1.0

𝜕2J󰜈(t) 𝜕󰜈 2

0.0

–1.0 –2.0 –3.0 0.0 0.2 0.4 t 0.6 0.8 1.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.23: Second derivatives with respect to the order of the Anger functions ∂2Jν(t)/∂ν2 as a function of ν and t.

2.3 First, Second and Third Derivatives with Respect to the Order

213

2.0 1.0

𝜕2J󰜈(t) 𝜕󰜈 2

0.0 –1.0 –2.0 –3.0

0.0 1.0 2.0 t 3.0 4.0 5.0 0.0

2.0

1.0

3.0

4.0

5.0

Figure 2.24: Second derivatives with respect to the order of the Anger functions ∂2Jν(t)/∂ν2 as a function of ν and t.

8.0 1

4.0

2 3

𝜕3J󰜈(t) 3 𝜕󰜈

6

0.0

4

5 –4.0

–8.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.25: Third derivatives of the Anger function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 5 – t = 5.0.

2 Numerical Results – Tabulation of First, Second and Third Derivatives

214

8.0 6.0 4.0

𝜕3J󰜈(t)

2.0

𝜕󰜈3

0.0

–2.0 –4.0

t

0.0 0.2 0.4 0.6 0.8 1.0

1.0

0.0

3.0

2.0

4.0

5.0

Figure 2.26: Third derivatives with respect to the order of the Anger functions ∂3Jν(t)/∂ν3 as a function of ν and t.

8.0 6.0 4.0

𝜕3J󰜈(t) 3 𝜕󰜈

2.0 0.0

–2.0 –4.0 0.0 1.0 2.0 t 3.0

4.0

5.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.27: Third derivatives with respect to the order of the Anger functions ∂3Jν(t)/∂ν3 as a function of ν and t.

2.3 First, Second and Third Derivatives with Respect to the Order

215

Table 2.7: First derivatives with respect to the order of the Anger functions ∂Jν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

216

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

2.3 First, Second and Third Derivatives with Respect to the Order

217

Table 2.7 (continued ) t ν . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

218

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.3 First, Second and Third Derivatives with Respect to the Order

219

Table 2.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

t ν . .

.

.

.

.

.

.

. .

. .

. .

. .

. .

. .

220

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . .

2.3 First, Second and Third Derivatives with Respect to the Order

221

Table 2.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

222

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.7 (continued ) t ν . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

2.3 First, Second and Third Derivatives with Respect to the Order

223

Table 2.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

224

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

225

2.3 First, Second and Third Derivatives with Respect to the Order

Table 2.8: Second derivatives with respect to the order of the Anger functions ∂2Jν(t)/∂ν2. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

226

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

2.3 First, Second and Third Derivatives with Respect to the Order

227

Table 2.8 (continued ) t ν . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. .

t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

228

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

2.3 First, Second and Third Derivatives with Respect to the Order

229

Table 2.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

t ν . .

.

.

.

.

.

.

−. −.

−. −.

−. −.

−. −.

−. −.

−. −.

230

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

2.3 First, Second and Third Derivatives with Respect to the Order

231

Table 2.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

232

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.8 (continued ) t ν . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. .

t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

2.3 First, Second and Third Derivatives with Respect to the Order

233

Table 2.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

234

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

2.3 First, Second and Third Derivatives with Respect to the Order

235

Table 2.9: Third derivatives with respect to the order of the Anger functions ∂3Jν(t)/∂ν3. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

236

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.3 First, Second and Third Derivatives with Respect to the Order

237

Table 2.9 (continued ) t ν . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. . . . . . . . . . . . .

−. −. −. −. . . . . . . . . . . . .

−. −. −. −. . . . . . . . . . . . .

−. −. −. −. . . . . . . . . . . . .

−. −. −. −. . . . . . . . . . . . .

−. −. −. −. . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. . . . . . . . . . . . . . . . . . . . . .

−. −. . . . . . . . . . . . . . . . . . . . . .

−. −. . . . . . . . . . . . . . . . . . . . . .

−. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. . . . . . . . . . . . . . . . . . . . .

238

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

2.3 First, Second and Third Derivatives with Respect to the Order

239

Table 2.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

t ν . . .

.

.

.

.

.

.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

240

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.3 First, Second and Third Derivatives with Respect to the Order

241

Table 2.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

242

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.9 (continued ) t ν . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

.

.

. . . . . . . . . . . .

.

.

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

2.3 First, Second and Third Derivatives with Respect to the Order

243

Table 2.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

244

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

2.4 First, Second and Third Derivatives with Respect to the Order

245

2.4 First, Second and Third Derivatives with Respect to the Order of the Weber Functions Derivatives with respect to the order of the Weber functions were determined using MATHEMATICA program by applying the following integral representations: ∂ Eν ðzÞ 1 =− ∂ν π

ðπ

∂ 2 Eν ðzÞ 1 =− ∂ν2 π ∂ 3 Eν ðzÞ 1 = ∂ν3 π

t cosðν t − z sin tÞ dt

(2:4:1)

0

ðπ t2 sinðνt − z sin tÞdt

(2:4:2)

0

ðπ t3 cosðνt − z sin tÞ dt

(2:4:3)

0

Oscillatory form of these derivatives is similar to that observed in the case of Anger functions.

1.0

1

6 0.0

𝜕E󰜈(t)

4

3

𝜕󰜈

5

–1.0

2 –2.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.28: First derivatives of the Weber function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

2 Numerical Results – Tabulation of First, Second and Third Derivatives

246

1.0 0.5 0.0

𝜕E󰜈(t) 𝜕󰜈

–0.5 –1.0 –1.5

t

0.0 0.2 0.4 0.6 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.29: First derivatives with respect to the order of the Weber functions ∂Eν(t)/∂ν as a function of ν and t.

1.0 0.5

𝜕E󰜈(t) 𝜕󰜈

0.0 –0.5 –1.0 –1.5 0.0 1.0 2.0 3.0 t 4.0 5.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.30: First derivatives with respect to the order of the Weber functions ∂Eν(t)/∂ν as a function of ν and t.

2.4 First, Second and Third Derivatives with Respect to the Order

2.0

247

4

1.0

6

5

0.0 𝜕2E󰜈(t) 𝜕󰜈

2

3

–1.0 –2.0

2

–3.0

1 –4.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.31: Second derivatives of the Weber function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

2.0 1.0

𝜕2E󰜈(t) 𝜕󰜈 2

0.0

–1.0 –2.0 –3.0 0.0 0.2 0.4 0.6 t 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.32: Second derivatives with respect to the order of the Weber functions ∂2Eν(t)/∂ν2 as a function of ν and t.

2 Numerical Results – Tabulation of First, Second and Third Derivatives

248

2.0 1.0 𝜕2E󰜈(t) 2 𝜕󰜈

0.0 –1.0 –2.0

0.0 1.0 2.0 3.0 t 4.0 5.0

0.0

1.0

3.0

2.0

4.0

5.0

Figure 2.33: Second derivatives with respect to the order of the Weber functions ∂2Eν(t)/∂ν2 as a function of ν and t.

8.0 3

1

4.0 4 𝜕3E󰜈(t) 𝜕󰜈 3

6

0.0

–4.0

–8.0 0.0

5 2

1.0

2.0

3.0

4.0

5.0

Figure 2.34: Third derivatives of the Weber function with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

2.4 First, Second and Third Derivatives with Respect to the Order

8.0

𝜕3E󰜈(t) 𝜕󰜈 3

4.0 0.0 –4.0

0.0 0.2 0.4 t 0.6

0.8 1.0

2.0

1.0

3.0

4.0

5.0

Figure 2.35: Third derivatives with respect to the order of the Weber functions ∂3Eν(t)/∂ν3 as a function of ν and t.

8.0

𝜕3E󰜈(t) 𝜕󰜈 3

4.0

0.0 –4.0

0.0 1.0 2.0 3.0 t 4.0 5.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 2.36: Third derivatives with respect to the order of the Weber functions ∂3Eν(t)/∂ν3 as a function of ν and t.

249

250

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.10: First derivatives with respect to the order of the Weber functions ∂Eν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −.

2.4 First, Second and Third Derivatives with Respect to the Order

251

Table 2.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

252

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.10 (continued ) t ν . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . .

2.4 First, Second and Third Derivatives with Respect to the Order

253

Table 2.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

254

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

t ν . . .

.

.

.

.

.

.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

2.4 First, Second and Third Derivatives with Respect to the Order

255

Table 2.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −.

256

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

2.4 First, Second and Third Derivatives with Respect to the Order

257

Table 2.10 (continued ) t ν . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. −. −. −. −. −. −. . .

.

.

t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

258

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.4 First, Second and Third Derivatives with Respect to the Order

259

Table 2.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

260

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.11: Second derivatives with respect to the order of the Weber functions ∂2Eν(t)/∂ν2. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . .

2.4 First, Second and Third Derivatives with Respect to the Order

261

Table 2.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

262

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.11 (continued ) t ν . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.4 First, Second and Third Derivatives with Respect to the Order

263

Table 2.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

264

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

.

.

.

t ν . . . .

. . . . .

. . . . .

. . . . .

. . . .

. . . .

. . . .

2.4 First, Second and Third Derivatives with Respect to the Order

265

Table 2.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

266

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

2.4 First, Second and Third Derivatives with Respect to the Order

267

Table 2.11 (continued ) t ν . . . . . . . . . . . . .

.

.

.

.

.

.

. . −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −.

.

.

t ν . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

268

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.4 First, Second and Third Derivatives with Respect to the Order

269

Table 2.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

270

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.12: Third derivatives with respect to the order of the Weber functions ∂3Eν(t)/∂ν3. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

2.4 First, Second and Third Derivatives with Respect to the Order

271

Table 2.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

272

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.12 (continued ) t ν . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . −. −.

t ν . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . −. −. −. −. −.

. . . . . . . . . . . . . . −. −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −.

2.4 First, Second and Third Derivatives with Respect to the Order

273

Table 2.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

274

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

2.4 First, Second and Third Derivatives with Respect to the Order

275

Table 2.12 (continued ) t ν . .

.

.

.

.

.

.

−. −.

−. −.

−. −.

−. −.

−. −.

−. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

276

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

2.4 First, Second and Third Derivatives with Respect to the Order

277

Table 2.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. . . . . . . . . . . . . . . . . . . . . −. −.

−. . . . . . . . . . . . . . . . . . . . . −. −.

−. . . . . . . . . . . . . . . . . . . . . −. −.

−. . . . . . . . . . . . . . . . . . . . . −. −.

−. . . . . . . . . . . . . . . . . . . . . −. −.

−. . . . . . . . . . . . . . . . . . . . . . −.

.

.

.

−. −. −. . . . . . . . . . . . .

−. −. −. . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . .

.

.

.

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

278

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . .

2.4 First, Second and Third Derivatives with Respect to the Order

279

Table 2.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

280

2 Numerical Results – Tabulation of First, Second and Third Derivatives

Table 2.12 (continued ) t ν . . . . .

.

.

.

.

.

.

. . . . −.

. . . . −.

. . . . −.

. . . . −.

. . . −. −.

. . . −. −.

3 Numerical Results – Tabulation of the Kelvin Functions, Their First and Second Derivatives with Respect to the Order The Kelvin functions ber(t), bei(t), ker(t) and kei(t) and their first derivatives with regard to argument t were tabulated by McLachlan in his book Bessel Functions for Engineers [22] in 0 ≤ ν ≤ 10 range with 0.10 intervals. For the same range of t, he also gave values of bern(t), bein(t), kern(t) and kein(t) with n = 1, 2, 3, 4 and 5. In MATHEMATICA program, the Kelvin functions are available for any argument t and order ν. Similarly as in cases of other functions, the Kelvin functions are presented here in 0 ≤ ν ≤ 5 range with 0.05 intervals, and arguments t are in 0.05 ≤ t ≤ 5 range with 0.05 intervals for t ≤ 1.0 and with 1.0 intervals for t > 1.0. The first and second derivatives with respect to the order of the Bessel functions were always calculated by using the central-difference formulas of order O(h4) with h = 0.001.

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives with Respect to the Order Applying the central-difference formulas of order O(h4) with h = 10−3, derivatives with respect to the order of the Kelvin function (Figures 3.1–3.18) of the first kind were evaluated using MATHEMATICA program (Tables 3.1–3.6): ∂ berν ðtÞ −berν+2 h ðtÞ + 8berν+ h ðtÞ − 8berν−h ðtÞ + berν−2 h ðtÞ = ∂ν 12 h

(3:1:1)

∂ 2 berν ðtÞ −berν+2 h ðtÞ + 16berν+h ðtÞ − 30 berν ðtÞ + 16berν−h ðtÞ − berν−2 h ðtÞ = ∂ ν2 12 h2

(3:1:2)

https://doi.org/10.1515/9783110682472-003

3 Numerical Results – Tabulation of the Kelvin Functions

282

6.0

3.0 2 ber󰜈(t)

0.0

5 1

3

4

–3.0

6 –6.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.1: The Kelvin functions berν(t) plotted as a function of the order ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

0.8

0.4 ber󰜈(t)

0.0 –0.4 0.0 0.2 0.4 t

0.6 0.8 1.0 0.0

1.0

2.0

3.0

4.0

Figure 3.2: The Kelvin functions berν(t) as a function of ν and t.

5.0

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

283

4.0 2.0 0.0 ber󰜈(t)

–2.0 –4.0 –6.0 0.0 1.0 2.0 3.0

t

4.0 5.0 0.0

2.0

1.0

4.0

3.0

5.0

Figure 3.3: The Kelvin functions berν(t) as a function of ν and t.

12.0

6 6.0

5

𝜕ber󰜈 (t) 𝜕󰜈

4 2

0.0

1 3 –6.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.4: First derivatives of the Kelvin functions berν(t) with respect to the order ν as a function of ν at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

284

3 Numerical Results – Tabulation of the Kelvin Functions

1.0

0.0 𝜕ber󰜈(t) 𝜕󰜈

–1.0 –2.0 –3.0 0.0 0.2 0.4 t

0.6 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.5: First derivatives with respect to the order of the Kelvin functions ∂berν(t)/∂ν as a function of ν and t.

10.0

5.0 𝜕ber󰜈(t) 𝜕󰜈

0.0

–5.0 0.0 1.0 t

2.0 3.0 4.0 5.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.6: First derivatives with respect to the order of the Kelvin functions ∂berν(t)/∂ν as a function of ν and t.

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

285

18.0 6 12.0

5

6.0 𝜕2ber󰜈(t) 2 𝜕󰜈

4

1 0.0

3

2

–6.0 –12.0 –18.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.7: Second derivatives of the Kelvin functions berν(t) with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

4.0

𝜕2ber󰜈(t)

0.0

𝜕𝜈2 –4.0 0.0 0.2 0.4 t 0.6 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.8: Second derivatives with respect to the order of the Kelvin functions ∂2berν(t)/∂ν2 as a function of ν and t.

3 Numerical Results – Tabulation of the Kelvin Functions

286

16.0

8.0 д2ber󰜈(t)

0.0

д𝜈2 –8.0 –16.0

t

0.0 1.0 2.0 3.0 4.0 5.0

1.0

0.0

3.0

2.0

4.0

5.0

󰜈

Figure 3.9: Second derivatives with respect to the order of the Kelvin functions ∂2berν(t)/∂ν2 as a function of ν and t.

Table 3.1: The Kelvin functions berν(t). t ν . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . −. −. −.

. . . . . . . . . . . . . . −. −. −.

. . . . . . . . . . . . . . −. −. −.

. . . . . . . . . . . . . . −. −. −.

. . . . . . . . . . . . . . −. −. −.

. . . . . . . . . . . . . . −. −. −.

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

287

Table 3.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

288

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

.

.

.

.

.

. . . . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

t ν . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

289

Table 3.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

290

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

291

Table 3.1 (continued ) t ν

.

. . . . . . .

. . . . . . . .

.

.

. . . . . . .

.

.

. . . . . . .

. . . . . . .

. . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

292

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

293

Table 3.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

t ν . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . −. −.

. . . . . . . . −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

294

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −.

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

295

Table 3.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

296

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.2: First derivatives with respect to the order of the Kelvin functions ∂berν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

297

Table 3.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

298

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.2 (continued ) t ν

.

.

. . . . . . . . . . . .

.

.

. . . . .

.

.

. . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

299

Table 3.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

300

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

t ν . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −.

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

301

Table 3.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −.

302

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

303

Table 3.2 (continued ) t ν . . . . .

.

.

.

.

.

.

. . . . −.

. . . . −.

. . . . −.

. . . . −.

. . . . −.

. . . . −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . .

−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

304

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . .

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . .

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . .

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

305

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

Table 3.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . −.

. . . . . . . . . . . . . . . . . . . . . . −.

. . . . . . . . . . . . . . . . . . . . . . −.

. . . . . . . . . . . . . . . . . . . . . . −.

. . . . . . . . . . . . . . . . . . . . −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −.

Table 3.3: Second derivatives with respect to the order of the Kelvin functions ∂2berν(t)/∂ν2. t ν . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . .

−. . . . . . . . . . . .

−. −. . . . . . . . . . .

−. −. −. . . . . . . . . .

−. −. −. −. . . . . . . . .

−. −. −. −. −. . . . . . . .

306

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . .

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

307

Table 3.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

308

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.3 (continued ) t ν

.

.

.

. .

.

.

.

−. −.

−. −.

−. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

309

Table 3.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

310

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. . . . . . . . . . . . .

−. −. −. −. −. −. −. −. . . . . . . . . . . . .

−. −. −. −. −. −. −. −. . . . . . . . . . . . .

−. −. −. −. −. −. −. −. . . . . . . . . . . . .

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

311

Table 3.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

312

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . .

.

.

.

.

.

.

−. −.

−. −.

−. −.

−. .

. .

. .

3.1 The Kelvin Functions berν(t), and Their First and Second Derivatives

313

Table 3.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

314

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

315

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

Table 3.3 (continued ) t ν . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives with Respect to the Order These derivatives with respect to the order were evaluated directly, by using the central-difference formulas of order O(h4) with h = 10−3. The Kelvin functions beiν(t) were taken from MATHEMATICA program: ∂ beiν ðtÞ −beiν+2 h ðtÞ + 8beiν+ h ðtÞ − 8beiν−h ðtÞ + beiν−2 h ðtÞ = ∂ν 12h

(3:2:1)

∂ 2 beiν ðtÞ −beiν+2 h ðtÞ + 16beiν+ h ðtÞ − 30beiν ðtÞ + 16beiν−h ðtÞ − beiν−2 h ðtÞ = ∂ ν2 12 h2

(3:2:2)

Functional form of berν(t) and beiν(t) functions and their derivatives with respect to the order is similar.

3 Numerical Results – Tabulation of the Kelvin Functions

316

6.0

3.0 3 bei󰜈(t) 0.0 1

2

4

5 –3.0 6 –6.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.10: The Kelvin functions beiν(t) plotted as a function of the order ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

0.8 0.6 0.4 bei󰜈(t)

0.2 0.0 0.0 0.2 0.4 t

0.6 0.8 1.0

0.0

1.0

2.0

3.0

4.0

Figure 3.11: The Kelvin functions beiν(t) as a function of ν and t.

5.0

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

317

4.0 2.0 0.0 bei󰜈(t) –2.0 –4.0 0.0 1.0 2.0 t

3.0 4.0 5.0

0.0

2.0

1.0

4.0

3.0

5.0

Figure 3.12: The Kelvin functions beiν(t) as a function of ν and t.

12.0 8.0 6 4.0 𝜕bei󰜈 (t) 𝜕𝜈

2

–4.0 –8.0 –12.0 0.0

3

1

0.0 4 5

1.0

2.0

3.0

4.0

5.0

Figure 3.13: First derivatives of the Kelvin functions beiν(t) with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

318

3 Numerical Results – Tabulation of the Kelvin Functions

1.5 1.0 0.5 𝜕bei󰜈(t) 𝜕𝜈

0.0 –0.5 –1.0 0.0 0.2 0.4 0.6

t

0.8 1.0

0.0

1.0

4.0

3.0

2.0

5.0

Figure 3.14: First derivatives with respect to the order of the Kelvin functions ∂beiν(t)/∂ν as a function of ν and t.

10.0

5.0 𝜕bei󰜈(t) 𝜕𝜈

0.0

–5.0

0.0 1.0 t

2.0 3.0 4.0 5.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.15: First derivatives with respect to the order of the Kelvin functions ∂beiν(t)/∂ν as a function of ν and t.

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

319

20.0

6 10.0

5 𝜕2ber󰜈(t) 2 𝜕󰜈

2

0.0

1 4

3 –10.0

–20.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.16: Second derivatives of the Kelvin functions beiν(t) with respect to the order ν as a function of ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 1.0; 3 – t = 2.0; 4 – t = 3.0; 5 – t = 4.0; 6 – t = 5.0.

2.0 0.0 –2.0 𝜕2bei󰜈(t) 2

𝜕󰜈

–4.0 –6.0 –8.0 0.0 0.2 0.4 t

0.6 0.8 1.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.17: Second derivatives with respect to the order of the Kelvin functions ∂2beiν(t)/∂ν2 as a function of ν and t.

3 Numerical Results – Tabulation of the Kelvin Functions

320

10.0 5.0 𝜕2bei󰜈(t)

0.0

𝜕󰜈2 –5.0 –10.0 –15.0 0.0 1.0 2.0 t

3.0 4.0 5.0

0.0

1.0

3.0

2.0

4.0

5.0

Figure 3.18: Second derivatives with respect to the order of the Kelvin functions ∂2beiν(t)/∂ν2 as a function of ν and t.

Table 3.4: The Kelvin functions beiν(t). t ν . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

321

Table 3.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

322

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

.

.

.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

t ν . . .

. . . .

. . . .

. . . .

. . .

. . .

. . .

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

323

Table 3.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

324

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

325

Table 3.4 (continued ) t ν . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

326

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

327

Table 3.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

. . . . . . . . . . .

. . . . . . . . . . −.

. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

328

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

329

Table 3.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

330

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.4 (continued ) t ν . . . .

.

.

.

.

.

.

−. −. −. −.

−. −. −. −.

−. −. −. −.

−. −. −. −.

−. −. −. −.

−. . . .

Table 3.5: First derivatives with respect to the order of the Kelvin functions ∂beiν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

331

Table 3.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

332

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

t ν . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . −. −. −. −.

. . . . . . . . . −. −. −.

. . . . . . . . . −. −. −.

. . . . . . . . . −. −. −.

. . . . . . . . . −. −. −.

. . . . . . . . . −. −. −.

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

333

Table 3.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

334

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

335

Table 3.5 (continued ) t ν . . .

. . . .

. . . .

.

.

. . .

.

.

. . .

. . .

. . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

336

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

337

Table 3.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. . . . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

338

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

339

Table 3.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . .

340

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.6: Second derivatives with respect to the order of the Kelvin functions ∂2beiν(t)/∂ν2. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

341

Table 3.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

342

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.6 (continued ) t ν

.

.

. . . . . . . . . . . . . . .

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . . −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

343

Table 3.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

344

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

t ν . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

345

Table 3.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

346

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 The Kelvin Functions beiν(t), and Their First and Second Derivatives

347

Table 3.6 (continued ) t ν . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

348

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

349

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

Table 3.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . −. −. −. −. −.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives with Respect to the Order The first and second derivatives with respect to the order were evaluated using the central-difference formulas of order O(h4) with h = 10−3, and the Kelvin functions kerν(t) were taken from MATHEMATICA program: ∂ kerν ðtÞ −kerν+2 h ðtÞ + 8kerν+ h ðtÞ − 8kerν−h ðtÞ + kerν−2 h ðtÞ = ∂ν 12 h

(3:3:1)

∂ 2 kerν ðtÞ −kerν+2 h ðtÞ + 16kerν+ h ðtÞ − 30kerν ðtÞ + 16kerν−h ðtÞ − kerν−2 h ðtÞ = ∂ν2 12 h2

(3:3:2)

3 Numerical Results – Tabulation of the Kelvin Functions

350

The behaviour of the Kelvin functions kerν(t) and its derivatives with respect to the order is complicated (Tables 3.7–3.9), and the functions have large positive and negative values which depend strongly on ν and t (Figures 3.19–3.24). 1,500 1,000 500 0

ker󰜈(t)

–500 0.0 0.2 0.4 0.6

t

0.8 1.0 0.0

1.0

2.0

4.0

3.0

5.0

Figure 3.19: The Kelvin functions kerν(t) as a function of ν and t.

3,000 2,000 1,000 ker󰜈(t)

0 –1,000 0.0 1.0 t

2.0 3.0 4.0 5.0

0.0

1.0

2.0

3.0

4.0

Figure 3.20: The Kelvin functions kerν(t) as a function of ν and t.

5.0

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

1,500 1,000 500 𝜕ker󰜈(t) 𝜕󰜈

0 –500 0.0 0.2 0.4 t 0.6 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.21: First derivatives with respect to the order of the Kelvin functions ∂kerν(t)/∂ν as a function of ν and t.

3,000

𝜕ker󰜈(t) 𝜕󰜈

2,000

1,000 0

0.0 1.0 2.0 3.0 t 4.0 5.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.22: First derivatives with respect to the order of the Kelvin functions ∂kerν(t)/∂ν as a function of ν and t.

351

352

3 Numerical Results – Tabulation of the Kelvin Functions

3,000 2,000 𝜕ker󰜈(t) 𝜕󰜈2

1,000 0 0.0

0.2

0.4 t

0.6

0.8

1.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.23: Second derivatives with respect to the order of the Kelvin functions ∂2kerν(t)/∂ν2 as a function of ν and t.

2,000

1,000 𝜕2ker󰜈(t) 𝜕󰜈 2

0 –1,000 0.0 1.0 2.0 t

3.0 4.0 5.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.24: Second derivatives with respect to the order of the Kelvin functions ∂2kerν(t)/∂ν2 as a function of ν and t.

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

353

Table 3.7: The Kelvin functions kerν(t). t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

.

.

.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

354

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. −,. −,. −, −, −, −, −, −.E+ −. −. −. −. −. −. −. −.E+ −. −. −.

. . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. −,. −,. −,. −,. −,. −,. −,. − −,, −,, −,, −,, −,, −,, − − − −.E+ −.

. . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. . −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. − − − − −,,

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

355

Table 3.7 (continued ) t ν . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −.E+ .E+ . .E+ . . . .

. −. −. −. −. −. −. −. −,, ,, .E+ . . .E+ . .

.

.

.

−,, −,, −,, −,, −,, −,, −,, −,. ,, ,,  .E+ . . .

−,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,, ,, ,, ,, ,,

−,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,. ,,. ,, ,, ,,

t ν . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

356

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

357

Table 3.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,. ,. ,. ,. ,,

−. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. ,. ,. ,. ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. ,. ,. ,. ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. −. . ,. ,. ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −. −. −. . ,. ,. ,. ,. ,. ,.

.

.

.

t ν . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

358

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . .

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

359

Table 3.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

360

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.7 (continued ) t ν . . . . . . .

. . . ,. ,. ,. ,. ,.

.

.

. . . ,. ,. ,. ,.

. . . . ,. ,. ,.

. . . . . . ,. ,.

. . . . . . . ,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

361

Table 3.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

362

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

t ν . . . . . . . . . . . . . .

.

.

.

.

.

. . −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. . . . .

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

363

Table 3.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

364

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

365

Table 3.8: First derivatives with respect to the order of the Kelvin functions ∂kerν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

366

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,,. ,,. −,. −,, −,, −,, −,, −.E+ −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −.E+ −. −.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. . −,. −,. −,. −,. −,. −,, −,, −,, −,, −,, −,, −,, −,, −,, −.E+ −. −. −. −. −. −. −. −. −.

. . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. . −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,v −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −,,

. . . . . . . . . . ,. ,. ,. ,. ,. ,. . . . −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,, −,, −,, −,, −,, −,, −,,

. . . . . . . . . . . . . . . . . . −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

367

Table 3.8 (continued ) t ν . . . . . . . . . . . . . .

. −. −. −. −.E+ .E+ . . . .E+ . . . . .

. −. −. − .E+ . .E+ . . . . . .E+ . .

.

.

.

−,, −,, ,,. ,, ,, .E+ . . . . .E+ . . .

−,. −,. ,. ,, ,, ,, ,, .E+ . . . . . .

−,. −,. ,. ,. ,, ,, ,,  ,v ,, ,, .E+ . .

.

.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

368

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −,. −,. −,.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −,.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

369

Table 3.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. . ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,,

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,,

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.  

−. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

−. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. −. . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

t ν . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

370

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

371

Table 3.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. −. . . ,. ,. ,. ,. ,. ,.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . ,. ,. ,. ,. ,. ,.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . ,. ,. ,. ,. ,.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . ,. ,. ,. ,.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . ,. ,. ,.

372

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.8 (continued ) t ν . . . . . .

. ,. ,. ,. ,. ,. ,.

.

.

,. ,. ,. ,. ,. ,.

,. ,. ,. ,. ,. ,.

. ,. ,. ,. ,. ,. ,.

. ,. ,. ,. ,. ,. ,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

373

Table 3.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

374

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . ,. ,. ,. ,. ,. ,. ,. ,.

.

.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . ,. ,. ,. ,. ,. ,. ,.

.

.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . ,. ,. ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . ,. ,.

.

.

t ν . . . . . . . . . . . . . .

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

375

Table 3.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

376

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

377

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

Table 3.9: Second derivatives with respect to the order of the Kelvin functions ∂2kerν(t)/∂ν2. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . ,. ,. ,. ,. ,. ,. ,. ,. ,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . ,. ,. ,. ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

378

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

,. ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,. −,,. −,, −,, −.E+ −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −.E+

,. ,. ,. ,. ,. ,. ,,. ,,. ,. ,. ,. ,. ,. ,. ,. −,. −,. −,. −,. − − − − − − − −.E+ −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. ,,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. −. −,. −,. −,. −,. −,. −,. −,. −,. − − − − − − − − − − − − − − − − − − −,. ,,

. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. . −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, ,, ,,

. . . . . . . . . . . . . . −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. ,. ,,

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

379

Table 3.9 (continued ) t ν . . . . . . . . . . . . . . .

. .E+ .E+ . . .E+ . . . . . .E+ . . . .

.

.

.E+ .E+ . . . . .E+ . . . . . . . .

.E+ . . . . .E+ . . . . . . . . .E+

. ,, ,, .E+ . . . . . . . .E+ . . . .

. ,, ,, ,, ,, ,, .E+ . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .

380

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,.

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

381

Table 3.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,. ,.         .E+ . . . . .

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,. ,. ,.             .E+

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. . ,. ,. ,. ,. ,. ,. ,. ,.          

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.        

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.    

t ν . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

−. −. −. −. −. −. −.

382

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

383

Table 3.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. −. −. . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

.

.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

384

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.9 (continued ) t ν . . . . . . .

. ,. ,. ,. ,. ,. ,. ,.

.

.

,. ,. ,. ,. ,. ,. ,.

,. ,. ,. ,. ,. ,. ,.

. ,. ,. ,. ,. ,. ,. ,.

. ,. ,. ,. ,. ,. ,. ,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

385

Table 3.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

386

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

.

.

−. −. −. . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

.

.

−. −. −. . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

−. −. −. . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

−. −. . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

.

.

.

t ν . . . . . . . . . . . . . .

. −. −. −. −. −. −. . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . −. −.

3.3 The Kelvin Functions kerν(t), and Their First and Second Derivatives

387

Table 3.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

388

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

389

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives with Respect to the Order The Kelvin functions keiν(t) (Tables 3.10–3.12) exist in MATHEMATICA program. Their first and second derivatives with respect to the order were determined directly using the central-difference formulas of order O(h4) with h = 10−3. The functional form of these functions resembles those of kerν(t) functions (Figures 3.25–3.30): ∂ keiν ðtÞ −keiν+2 h ðtÞ + 8keiν+ h ðtÞ − 8 keiν−h ðtÞ + keiν−2 h ðtÞ = ∂ν 12 h

(3:4:1)

∂ 2 keiν ðtÞ −keiν+2 h ðtÞ + 16keiν+ h ðtÞ − 30keiν ðtÞ + 16keiν−h ðtÞ − keiν−2 h ðtÞ = ∂ ν2 12 h2

(3:4:2)

800

400

0

kei󰜈(t)

–400

0.0 0.2 t

0.4 0.6 0.8 1.0

0.0

1.0

2.0

3.0

4.0

Figure 3.25: The Kelvin functions keiν(t) as a function of ν and t.

5.0

3 Numerical Results – Tabulation of the Kelvin Functions

390

3,000 2,000 1,000 kei󰜈(t)

0 –1,000 –2,000 0.0 1.0 2.0 3.0

t

4.0 5.0

1.0

0.0

4.0

3.0

2.0

5.0

Figure 3.26: The Kelvin functions keiν(t) as a function of ν and t.

800 400 𝜕kei󰜈(t) 𝜕󰜈

0 –400 –800 0.0

0.2

0.4 t

2.0

0.6

3.0

4.0

5.0

1.0

0.8 1.0

0.0

Figure 3.27: First derivatives with respect to the order of the Kelvin functions ∂keiν(t)/∂ν as a function of ν and t.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

391

4,000

2,000 𝜕kei󰜈(t)

0

𝜕󰜈

–2,000 0.0 1.0 2.0 t

3.0 4.0 5.0

2.0

1.0

0.0

3.0

4.0

5.0

Figure 3.28: First derivatives with respect to the order of the Kelvin functions ∂keiν(t)/∂ν as a function of ν and t.

1,000

500

𝜕2kei󰜈(t)

0

2

𝜕󰜈

–500

0.0 0.2 0.4 0.6 t 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 3.29: Second derivatives with respect to the order of the Kelvin functions ∂2keiν(t)/∂ν2 as a function of ν and t.

392

3 Numerical Results – Tabulation of the Kelvin Functions

3,000

2,000 𝜕2kei󰜈(t) 𝜕󰜈2

1,000 0 0.0 1.0 2.0 t 3.0

4.0 5.0

0.0

1.0

3.0

2.0

4.0

5.0

Figure 3.30: Second derivatives with respect to the order of the Kelvin functions ∂2keiν(t)/∂ν2 as a function of ν and t.

Table 3.10: The Kelvin functions keiν(t). t ν . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

393

Table 3.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. . −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −,. −,. −,. −,.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −,.

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

394

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −.E+ −. −. −. −. −. −,, ,. .E+ . . . .E+ . . . . . .E+ . . . . . . . . .

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. . ,. ,, ,, ,, ,, .E+ . . . . . . . .E+ . . . . . .

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. . ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,, .E+ . . . . . . .

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. . ,. ,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

−. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,,

t ν . .

.

.

.

.

.

−. −.

−. −.

−. −.

−. −.

−. −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

395

Table 3.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . .

396

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. −. . . ,. ,. ,. ,. ,. ,.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . ,. ,. ,. ,. ,. ,.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . ,. ,. ,. ,. ,.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . ,. ,. ,.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . ,. ,.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

397

Table 3.10 (continued ) t ν . . . . . . . . . . . . .

. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,, ,,

.

.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

398

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

399

Table 3.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . ,. ,. ,.

t ν . . . . .

.

.

.

.

.

−. −. −. −. −.

−. −. −. −. −.

−. −. −. −. −.

−. −. −. −. −.

−. −. −. −. −.

400

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

401

Table 3.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

402

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.10 (continued ) t ν

.

.

.

.

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

403

Table 3.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

404

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.10 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −.

Table 3.11: First derivatives with respect to the order of the Kelvin functions ∂keiν(t)/∂ν. t ν . . . . .

.

.

−. −. −. −. −.

−. −. −. −. −.

. −. −. −. −. −.

.

.

−. −. −. −. −.

−. −. −. −. −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

405

Table 3.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

406

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

,. . −,. −,. −,. −,. −,. −,. −,. − − − − − − − − −. + −. −. −. −. −. −. −. −. −. −. −,, .E+ . .E+ . . . . . .E+ . . . . . .

. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,, −,, −,, −,, −,, −,, −,, −,, −,, −, , ,, ,, ,, .E+ . . . . . . .E+ . . . .

−. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,. ,, ,, ,, ,, ,, ,, .E+ . . . . . .

−. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. . ,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

−. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,,

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

407

Table 3.11 (continued ) t ν . . . . . . . . .

. . .E+ . . . . . . .

.

.

. . . . . . . . .

. . . . . . . . .

. .E+ . . . . . . . .

. ,, ,, ,, ,, ,, ,, ,, ,, ,,

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . .

408

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −.

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

409

Table 3.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −,. −. . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

.

.

−. −. . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

.

.

−. . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

−. . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

−. . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

t ν . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. .

410

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

411

Table 3.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. .

412

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.11 (continued ) t ν . .

. ,. ,.

.

.

,. .

,. .

. . .

. . −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

413

Table 3.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . .

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

414

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. . . . −.

.

.

. . . . . . . . . . . . . . . . . . . −.

.

.

. . . . . . . . . . . . . . . . . . . −.

. . . . . . . . . . . . . . . . . . . −.

. . . . . . . . . . . . . . . . . . . −.

.

.

.

t ν . . . . . . . . . . . . . . . . . . .

. −. −. −. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . −. −. −. −. −. −. −. −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

415

Table 3.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

416

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.11 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

417

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

Table 3.12: Second derivatives with respect to the order of the Kelvin functions ∂2keiν(t)/∂ν2. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. . .

. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

.

.

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

−. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

418

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

,. −. −,. −,. −,. −,. −,. −,, −,, −,, −,, −,, −,, −,, −.E+ −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . .E+ . . . .E+ . . . . . . .E+ .

−. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −,, −,. ,, ,, ,, .E+ . . . . . .E+ . . . . . .

−. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,, ,, ,, ,, ,, ,, .E+ . . . . . . . .

−. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,, ,, .E+ . . .

−. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. ,. ,. ,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,, ,, ,,

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

419

Table 3.12 (continued ) t ν

.

. . . . . . . . . . . . .

. . . . . . . . . . . . −.

.

.

.

.

. . . . .E+ . . . . .E+ . −. −.E+

. .E+ . . . . . . . .E+ −. −.E+ −.

. . . . . . . . . ,, −.E+ −. −.

,, ,, ,, ,, ,, ,, ,, ,, ,, −,, −,, −.E+ −.

.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

420

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. . ,. ,.

.

.

.

.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −. −. . ,. ,.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −. −. −. . . ,. ,.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . ,. ,.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . ,.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

421

Table 3.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,. −,, −,, −,, −.E+

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,, ,, ,, ,, ,, ,, ,, ,, , , −,. −,. −,, −,, −,,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. −,. −,. −,, −,, −,,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. −,. −,. −,. −,, −,,

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. −. −,. −,. −,. −,. −,.

.

.

t ν . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

422

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

423

Table 3.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. . −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. −. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. . −. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. . −. −,. −,. −,.

424

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.12 (continued ) t ν . . .

.

.

.

.

.

−,. −,. −,.

−,. −,. −,.

−,. −,. −,.

−,. −,. −,.

−,. −,. −,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −.

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

425

Table 3.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . .

426

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. . . −. −,. −,. −,. −,. −,. −,.

. . . . ,. ,. ,. ,. ,. ,. ,. ,. . . . −. −. −,. −,. −,. −,. −,. −,.

. . . . . . . . . . . . . . . −. −. −,. −,. −,. −,. −,. −,.

. . . . . . . . . . . . . . . −. −. −. −,. −,. −,. −,. −,.

. . . . . . . . . . . . . . . −. −. −. −,. −,. −,. −,. −,.

.

.

.

.

. . . . . . . . . . . . . . .

. . . . . . . . . . . −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

3.4 The Kelvin Functions keiν(t), and Their First and Second Derivatives

427

Table 3.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

428

3 Numerical Results – Tabulation of the Kelvin Functions

Table 3.12 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4 Numerical Results – Tabulation of Functions, First and Second Derivatives with Respect to the Order of the Integral Bessel Functions The Bessel and Related Functions of the Same Argument and Order As already mentioned, the integral Bessel functions are tabulated in the Abramowitz– Stegun handbook [9], but only those Bessel functions that have zero order. Their integrands are (1 − J0(t))/t, (1 − I0(t))/t, Y0(t)/t and K0(t)/t, and the integration limits vary in 0 ≤ t ≤ 5 region in 0.1 intervals. The integral Bessel functions Jiν(t), Yiν(t) and Kiν(t) were calculated using the integration program of MATHEMATICA. They are presented here in 0 ≤ ν ≤ 5 range, with 0.05 intervals, and arguments t are in 0.05 ≤ t ≤ 5 range, with 0.05 intervals for t ≤ 1.0 and with 1.0 intervals for t > 1.0. The first and second derivatives with respect to the order of the integral Bessel functions were always calculated by using the central-difference formulas of order O(h4) with h = 0.001.

4.1 The Integral Bessel Functions of the First Kind and Their First and Second Derivatives with Respect to the Order Integral Bessel functions Jiν(t) are defined by ∞ ð

Jiν ðzÞ =

Jν ðtÞ dt t

(4:1:1)

z

They were calculated by applying MATHEMATICA program, and their derivatives with respect to the order of the integral Bessel function of the first kind were determined using the central – difference formulas: ∂ Jiν ðtÞ − Jiν + 2 h ðtÞ + 8Jiν + h ðtÞ − 8Jiν − h ðtÞ + Jiν − 2 h ðtÞ = ∂ν 12 h

(4:1:2)

∂ 2 Jiν ðtÞ − Jiν + 2 h ðtÞ + 16Jiν + h ðtÞ − 30 Jiν ðtÞ + 16 Jiν − h ðtÞ − Jiν − 2 h ðtÞ = ∂ ν2 12 h2

(4:1:3)

Changes in values of the integral Bessel functions and the first and second derivatives are important only for small values of the order ν (see Figures 4.1–4.9; Tables 4.1–4.3).

https://doi.org/10.1515/9783110682472-004

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

430

3.0

1

2.5 2.0

Ji𝜈 (t)

2

1.5

3

1.0 0.5

4

0.0 0.0

6

5 1.0

2.0

3.0

4.0

5.0

Figure 4.1: The integral Bessel functions of the first kind Jiν(t) plotted as a function of the order ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 0.25; 3 – t = 0.50; 4 – t = 0.75; 5 – t = 1.0; 6 – t = 1.5.

3.0 2.5 2.0

Ji𝜈(t)

1.5 1.0 0.5 0.0 0.0

0.2

0.4 t

0.6

0.8

1.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 4.2: The integral Bessel functions of the first kind Jiν(t) as a function of ν and t.

4.1 The Integral Bessel Functions of the First Kind

1.5 1.0

Jiv (t)

0.5 0.0 –0.5 0.0 1.0

2.0

t

3.0

4.0

1.0

2.0

3.0

4.0

5.0

5.0 0.0

Figure 4.3: The integral Bessel functions of the first kind Jiν(t) as a function of ν and t.

1.0 0.0 ∂Jiν(t)

–1.0

∂ν

4 6 3

5

2

–2.0 –3.0 –4.0 0.0

1 1.0

2.0

3.0

4.0

5.0

Figure 4.4: First derivatives of the integral Bessel functions of the first kind with respect to the order ∂Jiν(t)/∂ν plotted as a function of the order ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 0.25; 3 – t = 0.50; 4 – t = 0.75; 5 – t = 1.0; 6 – t = 1.5.

431

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

432

0.5 0.0 ∂Jiν(t) ∂ν

–0.5 –1.0 –1.5 –2.0 0.0 0.2 0.4 t 0.6 0.8 1.0

1.0

0.0

3.0

2.0

Figure 4.5: First derivatives of the integral Bessel functions of the first kind with respect to the order ∂Jiν(t)/∂ν as a function of ν and t.

1.0 0.5 0.0

∂Jiv (t) ∂v

–0.5 –1.0 –1.5 –2.0 0.0 1.0 2.0 t 3.0 4.0 5.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 4.6: First derivatives of the integral Bessel functions of the first kind with respect to the order ∂Jiν(t)/∂ν as a function of ν and t.

4.1 The Integral Bessel Functions of the First Kind

433

5.0 1

4.0 3.0 ∂2Jiν(t) 2.0 ∂ν2

1.0 0.0 –1.0

2

6

3

5

–2.0 0.0

4

1.0

2.0

3.0

4.0

5.0

Figure 4.7: Second derivatives of the integral Bessel functions of the first kind with respect to the order ∂2Jiν(t)/∂ν2 plotted as a function of the order ν, at constant values of argument t. 1 – t = 0.05; 2 – t = 0.25; 3 – t = 0.50; 4 – t = 0.75; 5 – t = 1.0; 6 – t = 1.5.

4.0

2.0

∂2Ji𝜈(t) ∂󰜈2

0.0

–2.0 0.0

0.2

0.4

t

0.6

0.8

1.0

5.0 4.0 3.0 2.0 1.0 0.0

Figure 4.8: Second derivatives of the integral Bessel functions of the first kind with respect to the order ∂2Jiν(t)/∂ν2 as a function of ν and t.

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

434

2.0 1.5 𝜕2Ji󰜈(t) 𝜕󰜈2

1.0 0.5 0.0 –0.5

0.0 1.0 2.0 3.0 t 4.0 5.0 0.0

1.0

3.0

2.0

4.0

Figure 4.9: Second derivatives of the integral Bessel functions of the first kind with respect to the order ∂2Jiν(t)/∂ν2 as a function of ν and t.

Table 4.1: The integral Bessel functions of the first kind Jiν(t). t ν . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

4.1 The Integral Bessel Functions of the First Kind

435

Table 4.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

436

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t ν . . .

.

.

.

.

.

.

. . .

. . .

. . .

. . .

. . .

. . .

4.1 The Integral Bessel Functions of the First Kind

437

Table 4.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

438

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 The Integral Bessel Functions of the First Kind

439

Table 4.1 (continued ) t ν . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

t

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

440

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.1 (continued ) t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 The Integral Bessel Functions of the First Kind

441

Table 4.1 (continued ) t

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t ν . . . . . . . . .

.

.

.

.

.

. . . . . . . . .

. . . . . . . . .

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −.

. . . . . . . . . .

442

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.1 The Integral Bessel Functions of the First Kind

443

Table 4.1 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

444

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.1 (continued ) t ν . . . . . .

.

.

. . . . . .

. . . . . .

.

.

. . . . . .

. . . . . .

. . . . . . .

. . . . . . .

Table 4.2: First derivatives with respect to the order of the integral Bessel functions of the first kind ∂Jiν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.1 The Integral Bessel Functions of the First Kind

445

Table 4.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

446

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

t ν . . . . . . . . . .

.

.

.

.

.

. . −. −. −. −. −. −. −. −.

. . . −. −. −. −. −. −. −.

. . . . . −. −. −. −. −.

. . . . . . . −. −. −.

. . . . . . . . −. −.

. . . . . . . . . .

4.1 The Integral Bessel Functions of the First Kind

447

Table 4.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

448

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.1 The Integral Bessel Functions of the First Kind

449

Table 4.2 (continued ) t ν . . . . .

.

.

.

.

.

.

−. −. −. −. −.

−. −. −. −. −.

−. −. −. −. −.

−. −. −. −. −.

−. −. −. −. −.

−. −. −. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

450

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.1 The Integral Bessel Functions of the First Kind

451

Table 4.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

−. −. −. −. . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

452

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . .

4.1 The Integral Bessel Functions of the First Kind

453

Table 4.2 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

454

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.3: Second derivatives with respect to the order of the integral Bessel functions of the first kind ∂ 2Jiν(t)/∂ν2. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 The Integral Bessel Functions of the First Kind

455

Table 4.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . −. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

456

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.3 (continued ) t ν . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . −. . . . . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.1 The Integral Bessel Functions of the First Kind

457

Table 4.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

458

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . −. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . −. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . −. . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . −. . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . −. . . . . . . . . . .

t ν . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

−. −. −. −. −. −.

4.1 The Integral Bessel Functions of the First Kind

459

Table 4.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . .

460

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1 The Integral Bessel Functions of the First Kind

461

Table 4.3 (continued ) t ν . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

.

.

. . . . . . . . .

.

.

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

.

.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

462

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

463

Table 4.3 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

−. −. . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind and Their First and Second Derivatives with Respect to the Order Integral Bessel functions Yiν(t) were calculated by applying MATHEMATICA program from their definition: ∞ ð

Yiν ðzÞ =

Yiν ðtÞ dt t

(4:2:1)

z

Their derivatives with respect to the order of the integral Bessel function of the second kind were determined using the following expressions (Figures 4.10–4.12; Tables 4.4–4.6):

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

464

∂ Yiν ðtÞ − Yiν + 2 h ðtÞ + 8 Yiν + h ðtÞ − 8Yiν − h ðtÞ + Yiν − 2 h ðtÞ = ∂ν 12 h

(4:2:2)

∂ 2 Yiν ðtÞ − Yiν + 2 h ðtÞ + 16Yiν + h ðtÞ − 30 Yiν ðtÞ + 16Yiν − h ðtÞ − Yiν − 2 h ðtÞ = ∂ν2 12 h2

(4:2:3)

0

–500

Yi󰜈(t) –1,000 –1,500

0.0 0.2 0.4 0.6 t 0.8 1.0 0.0

1.0

4.0

3.0

2.0

5.0

Figure 4.10: The integral Bessel functions of the second kind Yiν(t) as a function of ν and t.

0 –250

𝜕Yi󰜈(t) 𝜕󰜈

–500 –750 0.0 0.2 0.4 0.6 t 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 4.11: First derivatives of the integral Bessel functions of the second kind with respect to the order ∂Yiν(t)/∂ν as a function of ν and t.

4.2 Integral Bessel Functions of the Second Kind

465

0

–1,000 𝜕2Yi󰡶(t) 𝜕󰡶2

–2,000

0.0 0.2 0.4 0.6 t 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 4.12: Second derivatives of the integral Bessel functions of the second kind with respect to the order ∂2Yiν(t)/∂ν2 as a function of ν and t.

Table 4.4: The integral Bessel functions of the second kind Yiν(t). t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

466

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

467

Table 4.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. − − − − − − − − − − −.E+ −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −. −.E+ −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. − − − − − − − − − − − −.E+ −. −. −. −. −. −. −.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. − − − − − − − − − −

−. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. − − −

−. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

.

.

t ν . . . .

.

.

.

−. −. −. −.

. −. −. −.

. . −. −.

. . . .

. . . .

468

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

469

Table 4.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

470

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.4 (continued ) t ν . . . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −,. −,. −,. −,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

471

Table 4.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

472

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

t ν . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

4.2 Integral Bessel Functions of the Second Kind

473

Table 4.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

474

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

475

Table 4.4 (continued ) t ν . . . .

.

.

.

.

.

−. −. −. −.

−. −. −. −.

−. −. −. −.

−. −. −. −.

−. −. −. −.

.

.

.

−. −. −. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

476

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Integral Bessel Functions of the Second Kind

477

Table 4.4 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . −. −. −. −.

Table 4.5: First derivatives with respect to the order of the integral Bessel functions of the second kind ∂Yiν(t)/∂ν. t ν . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −.

478

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

479

Table 4.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. − − − − − − − − − − −.E+ −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,.

480

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.5 (continued ) t ν

.

. . . .

−. −. −. −.

. − − − −

.

.

.

−,. −,. −,. −,.

−,. −,. −,. −,.

−,. −,. −,. −,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

481

Table 4.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

482

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

t ν . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

483

Table 4.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

484

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

485

Table 4.5 (continued ) t ν . . .

.

.

.

.

.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

−. −. −.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

486

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

487

Table 4.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

.

t ν . . . . . . . . . . . . . . . . .

.

.

. . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. . . . . . .

488

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Integral Bessel Functions of the Second Kind

489

Table 4.5 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

490

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.6: Second derivatives with respect to the order of the integral Bessel functions of the second kind ∂2Yiν(t)/∂ν2. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

491

Table 4.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. − − − − − − − − − − −.E+ −. −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −. −. −. −.

−. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. − − − − − − − − − − − −.E+ −. −. −. −. −.

−. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. − − − − − − − −

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

492

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.6 (continued ) t ν . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −.E+ −. −. −. −. −. −. −. −. −.E+ −. −. −. −. −.

−. −. −. −. −. −.E+ −. −. −. −. −. −. −. −. −. −.

− − − − −.E+ −. −. −. −. −. −. −. −. −. −. −.

− − − − − − − − − − − − −.E+ −. −. −.

−,. −,. −,. −,. −,. − − − − − − − − − − −

t ν . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

493

Table 4.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

494

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. − − − − −

−. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,. −,.

t ν . . . . . .

.

.

.

.

.

−. . . . . .

−. −. −. . . .

−. −. −. −. . .

−. −. −. −. −. −.

−. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

495

Table 4.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

496

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −,. −,.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

497

Table 4.6 (continued ) t ν . . . . . . . . .

.

.

.

.

.

−,. −,. −,. −,. −,. −,. −,. −,. −,.

−,. −,. −,. −,. −,. −,. −,. −,. −,.

−. −,. −,. −,. −,. −,. −,. −,. −,.

−. −. −. −. −,. −,. −,. −,. −,.

−. −. −. −. −. −. −. −,. −,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . −. −. −. −. −. −. −. −.

498

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

499

Table 4.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

.

.

t ν . . . . . . . . . . . . .

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −.

. . −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . .

. . . . . . . . . . . . .

500

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4.2 Integral Bessel Functions of the Second Kind

501

Table 4.6 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

502

4.3 The Integral Modified Bessel Functions of the Second Kind and Their First and Second Derivatives with Respect to the Order Integral modified Bessel functions Kiν(t) are defined by ∞ ð

Kiν ðzÞ =

Kiν ðtÞ dt t

(4:3:1)

z

and they were calculated directly by applying MATHEMATICA program (Figures 4.13– Figures 4.15; Tables 4.7–4.9). First and second derivatives with respect to the order of the integral modified Bessel function of the second kind were determined using the following expressions: ∂ Kiν ðtÞ − Kiν + 2 h ðtÞ + 8Kiν + h ðtÞ − 8Kiν − h ðtÞ + Kiν − 2 h ðtÞ = ∂ν 12h

(4:3:2)

∂ 2 Kiν ðtÞ − Kiν + 2 h ðtÞ + 16Kiν + h ðtÞ − 30Kiν ðtÞ + 16Kiν − h ðtÞ − Kiν − 2 h ðtÞ = ∂ν2 12 h2

(4:3:3)

1×106 8×105 6×105 Ki𝜈 (t)

4×105 2×105 0 0.0 0.2 0.4 0.6 t 0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 4.13: The integral modified Bessel functions of the second kind Kiν(t) as a function of ν and t.

4.3 The Integral Modified Bessel Functions of the Second Kind

503

1×106 8×105 𝜕Ki󰣍 (t) 𝜕󰣍

6×105 4×105 2×105 0 0.0 0.2 0.4 t 0.6 0.8 1.0

1.0

0.0

2.0

3.0

4.0

5.0

Figure 4.14: First derivatives of the integral modified Bessel functions of the second kind with respect to the order ∂Kiν(t)/∂ν as a function of ν and t.

1×106 8×105 ∂2Kiv (t) ∂v2

6×105 4×105 2×105 0 0.0

0.2

0.4 t

0.6

0.8 1.0

0.0

1.0

2.0

3.0

4.0

5.0

Figure 4.15: Second derivatives of the integral modified Bessel functions of the second kind with respect to the order ∂2Kiν(t)/∂ν2 as a function of ν and t.

504

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.7: The integral modified Bessel functions of the second kind Kiν(t). t ν

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

505

Table 4.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.           .E+ . . . . . . .

. . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.      

. . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

506

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.7 (continued ) t ν . . . . . . . . . . . . . . .

. . .E+ . . . . . . . . . .E+ . . .

.

.

.

.

     .E+ . . . . . . . . .

,. ,.             .E+

,. ,. ,. ,. ,. ,. ,. ,. ,. ,.     

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

507

Table 4.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

508

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,.

t ν . . . . . . .

.

.

.

.

.

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

509

Table 4.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

510

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

511

Table 4.7 (continued ) t ν . . . . . . . .

.

.

.

.

.

. . . . ,. ,. ,. ,.

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

512

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

513

Table 4.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . .

t ν . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

514

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

515

Table 4.7 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

516

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.7 (continued ) t ν . . .

.

.

.

.

.

. . .

. . .

. . .

. . .

. . .

Table 4.8: First derivatives with respect to the order of the integral modified Bessel functions of the second kind ∂Kiν(t)/∂ν. t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

517

Table 4.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

. . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.           .E+ . .

. . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. 

. . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,.

518

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . .E+ . . . . . . . . .E+ . . . . . . . . . .E+

          .E+ . . . . . . . . . . .E+ . . . . .

,. ,. ,. ,. ,. ,. ,.              .E+ . . . . . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.            

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.      

t ν . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

519

Table 4.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

520

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

4.3 The Integral Modified Bessel Functions of the Second Kind

521

Table 4.8 (continued ) t ν . . .

.

.

.

.

.

,. ,. 

,. ,. ,.

,. ,. ,.

,. ,. ,.

,. ,. ,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

522

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

523

Table 4.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . ,.

. . . . . . . . . . . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

524

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

525

Table 4.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

526

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.8 (continued ) t ν . .

.

.

.

.

.

. .

. .

. .

. .

. .

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

527

Table 4.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

528

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.8 (continued ) t ν . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

Table 4.9: Second derivatives with respect to the order of the integral modified Bessel functions of the second kind ∂2Kiν(t)/∂ν2. t ν . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

529

Table 4.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.  

. . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

530

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

         .E+ . . . . . . . . .E+ . . . . . . . . . .E+ . . . . . . . . .E+ . . . . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,.            .E+ . . . . . . . . . . .E+ . . . . . . . . . .

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.              .E+ . . . . . . . . . . .E+

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.              .E+ . . . .

. . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.           

4.3 The Integral Modified Bessel Functions of the Second Kind

531

Table 4.9 (continued ) t ν . .

. . .

.

.

.E+ .

. .

. . .

.  

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

532

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

533

Table 4.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.        

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.   

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

t ν . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

534

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

535

Table 4.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,. ,. ,. ,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. ,.

536

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.9 (continued ) t ν . . .

.

.

.

.

.

,. ,. ,.

,. ,. ,.

,. ,. ,.

,. ,. ,.

,. ,. ,.

t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

537

Table 4.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

538

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . ,. ,.

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

t ν . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

4.3 The Integral Modified Bessel Functions of the Second Kind

539

Table 4.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

540

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.9 (continued ) t ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 The Bessel and Related Functions with the Same Argument and Order

541

4.4 The Bessel and Related Functions with the Same Argument and Order There are no tabulations of the Bessel and related functions with the same argument and order in the literature. Evidently, such values, but only in limited range, are available in tables of considered functions. Here, in systematic way, they are presented for the first time. The following functions Jν(ν), Yν(ν), Iν(ν), Kν(ν), Hν(ν), Lν(ν), Jν(ν), Eν(ν), berν(ν), beiν(ν), kerν(ν), keiν(ν), Jiν(ν), Yiν(ν) and Kiν(ν).are tabulated and plotted in figures. Usually, these functions are monotonic descending or rising curves (Figures 4.16, 4.17 and 4.19), or they have an oscillatory character (Figures 4.17 and 4.18).

4.0

4

2.0

3

1

0.0 2

–2.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 4.16: Functions with the same argument and order 1 – Jν(ν); 2 – Yν(ν); 3 – Iν(ν); 4 – Kν(ν).

2.0 2

1.5

1.0 3 1

0.5 4

0.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 4.17: Functions with the same argument and order 1 – Hν(ν); 2 – Lν(ν); 3 – Jν(ν); 4 – Eν(ν).

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

542

2.0 3

1.0 1

0.0 2

–1.0

4

0.0

1.0

2.0

3.0

4.0

5.0

Figure 4.18: Functions with the same argument and order 1 – berν(ν); 2 – beiν(ν); 3 – kerν(ν); 4 – keiν(ν).

4.0 3

2.0

1

0.0 2

–2.0 0.0

1.0

2.0

3.0

4.0

5.0

Figure 4.19: Functions with the same argument and order 1 – Jiν(ν); 2 – Yiν(ν); 3 – Kiν(ν).

543

4.4 The Bessel and Related Functions with the Same Argument and Order

Table 4.10: The Bessel functions with the same argument and order Jν(ν), Yν(ν), Iν(ν) and Kν(ν). ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jν(ν)

Yν(ν)

Iν(ν)

Kν(ν)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

544

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.10 (continued ) ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Jν(ν)

Yν(ν)

Iν(ν)

Kν(ν)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 The Bessel and Related Functions with the Same Argument and Order

545

Table 4.10 (continued ) ν . . . . . . . . . . . . .

Jν(ν)

Yν(ν)

Iν(ν)

Kν(ν)

. . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . .

. . . . . . . . . . . . .

Table 4.11: The Struve, Anger and Weber functions with the same argument and order Hν(ν), Lν(ν), Jν(ν) and Eν(ν). ν . . . . . . . . . . . . . . . . . . . . . . .

Hν(ν)

Lν(ν)

Jν(ν)

Eν(ν)

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

546

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.11 (continued ) ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hν(ν)

Lν(ν)

Jν(ν)

Eν(ν)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 The Bessel and Related Functions with the Same Argument and Order

547

Table 4.11 (continued ) ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Hν(ν)

Lν(ν)

Jν(ν)

Eν(ν)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

548

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.12: The Kelvin functions with the same argument and order berν(ν), beiν(ν), kerν(ν) and keiν(ν). ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

berν(ν)

beiν(ν)

kerν(ν)

keiν(ν)

. . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . .

4.4 The Bessel and Related Functions with the Same Argument and Order

549

Table 4.12 (continued ) ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

berν(ν)

beiν(ν)

kerν(ν)

keiν(ν)

. . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . .

−. −. −. −. −. −. . . . . . . . . . . . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −.

. . . . . . . . . . . . . . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . .

. . . −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . . . . . . . . . . . . . . .

550

4 Numerical Results – Tabulation of Functions, First and Second Derivatives

Table 4.12 (continued ) ν . . . . . . . . . . . .

berν(ν)

beiν(ν)

. . . . . . . . . . . .

kerν(ν)

−. −. −. −. −. −. −. −. −. −. . .

. . . . . . . . . . . .

keiν(ν) . . . . . . . −. −. −. −. −.

Table 4.13: The integral Bessel functions with the same argument and order Jiν(ν), Yiν(ν) and Kiν(ν). ν . . . . . . . . . . . . . . . . . . . . . . . . . .

Jiν(ν)

Yiν(ν)

Kiν(ν)

. . . . . . . . . . . . . . . . . . . . . . . . . .

−. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. −. . . . . −.

. . . . . . . . . . . . . . . . . . . . . . . . .

Index Anger function – first derivatives with respect to order 210, 211, 215–224 – functions of equal order and argument 210, 212, 213, 545–547 – integral representation 141, 210 – second derivatives with respect to order 210, 212, 213, 225–234 – third derivatives with respect to order 210, 213, 214, 235–244 Bessel function of the first kind – first derivatives with respect to order 1–3, 5–15 – functions of equal order and argument 2–4, 543–545 – integral representation 2 – second derivatives with respect to order 1–4, 15–25 – third derivatives with respect to order 1–5, 25–35 Bessel function of the second kind – central-difference formula approximation 35 – first derivatives with respect to the order 35–49 – functions of equal order and argument 543–545 – second derivatives with respect to the order 35, 36, 49–61 – third derivatives with respect to the order 35, 37, 62–74 Integral Bessel function of the first kind – central-difference formula approximation 429 – first derivatives with respect to order 429, 431, 432, 444–453 – functions of equal order and argument 430, 431, 433, 550 – second derivatives with respect to order 429, 433, 434, 454–463 – tabulation of the function 434–444 Integral Bessel function of the second kind – central-difference formula approximation 463

https://doi.org/10.1515/9783110682472-005

– first derivatives with respect to order 463, 477–489 – functions of equal order and argument 541–542, 550 – second derivatives with respect to order 463, 465, 490–501 – tabulation of the function 465–477 Kelvin function of the first kind – central-difference formula approximation 281 – first derivatives with respect to order 281, 283–281 – functions of equal order and argument 541–542, 548–550 – second derivatives with respect to order 281, 284–286 – tabulation of the function 286–315 Kelvin function of the second kind – central-difference formula approximation 315 – first derivatives with respect to order 315, 317–318 – functions of equal order and argument 541–542, 548–550 – second derivatives with respect to order 315, 319–320 – tabulation of the function 320–349 Modified Bessel function of the first kind – first derivatives with respect to order 74–76, 79–89 – functions of equal order and argument 74–76, 78 – integral representation 74 – second derivatives with respect to order 74, 76, 77, 89–98 – third derivatives with respect to order 74, 78, 79, 99–108 Modified Bessel function of the second kind – first derivatives with respect to order 109, 111–120 – functions of equal order and argument 109 – integral representation 109

552

Index

– second derivatives with respect to the order 109, 110, 120–130 – third derivatives with respect to the order 109, 110, 130–140 Modified integral Bessel function of the second kind – central-difference formula approximation 502 – first derivatives with respect to order 502, 503, 516–528 – functions of equal order and argument 541–542, 550 – second derivatives with respect to order 502, 503, 528–540 – tabulation of the function 504–516 Modified Kelvin function of the first kind – central-difference formula approximation 349 – first derivatives with respect to the order 349, 351–352 – functions of equal order and argument 541–542, 548–550 – second derivatives with respect to the order 349,352 – tabulation of the function 353–388 Modified Kelvin function of the second kind – central-difference formula approximation 389 – first derivatives with respect to the order 389, 390–391 – functions of equal order and argument 541–542, 548–550 – second derivatives with respect to the order 389, 391–392 – tabulation of the function 392–428

Modified Struve function – central-difference formula approximation 175 – first derivatives with respect to the order 175, 177, 180–189 – functions of equal order and argument 176, 177, 179 – second derivatives with respect to the order 175, 177, 190–199 – third derivatives with respect to the order 175, 179, 180, 200–209 Struve function – central-difference formula approximation 141 – first derivatives with respect to the order 141–143, 146–155 – functions of equal order and argument 141–143, 145, 545–547 – second derivatives with respect to the order 141, 143, 144, 156–165 – third derivatives with respect to the order 141, 145, 146, 165–175 Weber function – first derivatives with respect to the order 245, 246, 250–259 – functions of equal order and argument 245, 247, 248, 545–547 – integral representation 245 – second derivatives with respect to the order 245, 247, 248, 260–269 – third derivatives with respect to the order 245, 248, 249, 270–280