Handbook of Mathematical Functions [9 ed.] 0486612724

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Preface’ The present volume is an outgrowth of a Conference on Mathematical Tables held at Cambridge, Mass., on September 15-16, 1954, under the auspices of the National Science Foundation and the Massachusetts Institute of Technology. The purpose of the meeting was to evaluate the need for mathematical tables in the light of the availability of large scale computing machines. It was the consensus of opinion that in spite of the increasing use of the new machines the basic need for tables would continue to exist. Numerical tables of mathematical functions are in continual demand by scientists and engineers. A greater variety of functions and higher accuracy of tabulation are now required as a result of scientific advances and, especially, of the increasing use of automatic computers. In the latter connection, the tables serve mainly forpreliminarysurveys of problems before programming for machine operation. For those without easy access to machines, such tables are, of course, indispensable. Consequently, the Conference recognized that there was a pressing need for a modernized version of the classical tables of functions of Jahnke-Emde. To implement the project, the National Science Foundation requested the National Bureau of Standards to prepare such a volume and established an Ad Hoc Advisory Committee, with Professor Philip M. Morse of the Massachusetts Institute of Technology as chairman, to advise the staff of the National Bureau of Standards during the ~course of its preparation. In addition to the Chairman, the Committee consisted of A. Erdelyi, M. C. Gray, N. Metropolis, J. B. Rosser, H. C. Thacher, Jr., John Todd, C. B. Tompkins, and J. W. Tukey. The primary aim has been to include a maximum of useful information within the limits of a moderately large volume, with particular attention to the needs of scientists in all fields. An attempt has been made to cover the entire field of special functions. To carry out the goal set forth by tbe Ad Hoc Committee, it has been necessary to supplement the tables by including the mathematical properties that are important in computation work, as well as by providing numerical methods which demonstrate the use and extension of the tables. The Handbook was prepared under the direction of the late Milton Abramowitz, Its success has depended greatly upon the cooperation of and Irene A. Stegun. Their efforts together with the cooperation of the Ad HOC many mathematicians. Committee are greatly appreciated. The particular contributions of these and other individuals are acknowledged at appropriate places in the text. The sponsorship of the National Science Foundation for the preparation of the material is gratefully recognized. It is hoped that this volume will not only meet the needs of all table users but will in many cases acquaint its users with new functions. ALLEN V. ASTIN, L?imctor. Washington,

D.C.

Preface

to the Ninth

Printing

The enthusiastic reception accorded the “Handbook of Mathematical Functions” is little short of unprecedented in the long history of mathematical tables that began when John Napier published his tables of logarithms in 1614. Only four and one-half years after the first copy came from the press in 1964, Myron Tribus, the Assistant Secretary of Commerce for Science and Technology, presented the 100,OOOth copy of the Handbook to Lee A. DuBridge, then Science Advisor to the President. Today, total distribution is approaching the 150,000 mark at a scarcely diminished rate. The successof the Handbook has not ended our interest in the subject. On the contrary, we continue our close watch over the growing and changing world of computation and to discuss with outside experts and among ourselves the various proposals for possible extension or supplementation of the formulas, methods and tables that make up the Handbook. In keeping with previous policy, a number of errors discovered since the last printing have been corrected. Aside from this, the mathematical tables and accompanying text are unaltered. However, some noteworthy changes have been made in Chapter 2: Physical Constants and Conversion Factors, pp. 6-8. The table on page 7 has been revised to give the values of physical constants obtained in a recent reevaluation; and pages 6 and 8 have been modified to reflect changes in definition and nomenclature of physical units and in the values adopted for the acceleration due to gravity in the revised Potsdam system. The record of continuing acceptance of the Handbook, the praise that has come from all quarters, and the fact that it is one of the most-quoted scientific publications in recent years are evidence that the hope expressed by Dr. Astin in his Preface is being amply fulfilled. LEWIS M. BRANSCOMB,

Director

National Bureau of Standards November 1970

Foreword This volume is the result of the cooperative effort of many persons and a number of organizations. The National Bureau of Standards has long been turning out mathematical tables and has had under consideration, for at least IO years, the production of a compendium like the present one. During a Conference on Tables, called by the NBS Applied Mathematics Division on May 15, 19.52, Dr. Abramowitz of t,hat Division mentioned preliminary plans for such an undertaking, but indicated the need for technical advice and financial support. The Mathematics Division of the National Research Council has also had an active interest in tables; since 1943 it has published the quarterly journal, “Mathematical Tables and Aids to Computation” (MTAC),, editorial supervision being exercised by a Committee of the Division. Subsequent to the NBS Conference on Tables in 1952 the attention of the National Science Foundation was drawn to the desirability of financing activity in table production. With its support a z-day Conference on Tables was called at the Massachusetts Institute of Technology on September 15-16, 1954, to discuss the needs for tables of various kinds. Twenty-eight persons attended, representing scientists and engineers using tables as well as table producers. This conference reached consensus on several cpnclusions and recomlmendations, which were set forth in tbe published Report of the Conference. There was general agreement, for example, “that the advent of high-speed cornputting equipment changed the task of table making but definitely did not remove the need for tables”. It was also agreed that “an outstanding need is for a Handbook of Tables for the Occasional Computer, with tables of usually encountered functions and a set of formulas and tables for interpolation and other techniques useful to the occasional computer”. The Report suggested that the NBS undertake the production of such a Handbook and that the NSF contribute financial assistance. The Conference elected, from its participants, the following Committee: P. M. Morse (Chairman), M. Abramowitz, J. H. Curtiss, R. W. Hamming, D. H. Lehmer, C. B. Tompkins, J. W. Tukey, to help implement these and other recommendations. The Bureau of Standards undertook to produce the recommended tables and the To provide technical guidance National Science Foundation made funds available. to the Mathematics Division of the Bureau, which carried out the work, and to provide the NSF with independent judgments on grants ffor the work, the Conference Committee was reconstituted as the Committee on Revision of Mathematical Tables of the Mathematics Division of the National Research Council. This, after some changes of membership, became the Committee which is signing this Foreword. The present volume is evidence that Conferences can sometimes reach conclusions and that their recommendations sometimes get acted on. V

,/”

VI

FOREWORD

Active work was started at the Bureau in 1956. The overall plan, the selection of authors for the various chapters, and the enthusiasm required to begin the task were contributions of Dr. Abramowitz. Since his untimely death, the effort has continued under the general direction of Irene A. Stegun. The workers at the Bureau and the members of the Committee have had many discussions about content, style and layout. Though many details have had t’o be argued out as they came up, the basic specifications of the volume have remained the same as were outlined by the Massachusetts Institute of Technology Conference of 1954. The Committee wishes here to register its commendation of the magnitude and quality of the task carried out by the staff of the NBS Computing Section and their expert collaborators in planning, collecting and editing these Tables, and its appreciation of the willingness with which its various suggestions were incorporated into the plans. We hope this resulting volume will be judged by its users to be a worthy memorial to the vision and industry of its chief architect, Milton Abramowitz. We regret he did not live to see its publication. P. M. MORSE, Chairman. A. ERD~LYI M. C. GRAY N. C. METROPOLIS J. B. ROSSER H. C. THACHER. Jr. JOHN TODD ‘C. B. TOMPKINS J. W. TUKEY.

Handbook

of Mathematical

Functions

with

Formulas, Edited

Graphs, by Milton

1.

and Mathematical Abramowitz

and Irene A. Stegun

Introduction

The present Handbook has been designed to provide scientific investigators with a comprehensive and self-contained summary of the mathematical functions that arise in physical and engineering problems. The well-known Tables of Funct.ions by E. Jahnke and F. Emde has been invaluable to workers in these fields in its many editions’ during the past half-century. The present volume ext,ends the work of these authors by giving more extensive and more accurate numerical tables, and by giving larger collections of mathematical properties of the tabulated functions. The number of functions covered has also been increased. The classification of functions and organization of the chapters in this Handbook is similar to that of An Index of Mathematical Tables by A. Fletcher, J. C. P. Miller, and L. Rosenhead. In general, the chapters contain numerical tables, graphs, polynomial or rational approximations for automatic computers, and statements of the principal mathematical properties of the tabulated functions, particularly those of computa-

2.

Tables

Accuracy

The number of significant figures given in each table has depended to some extent on the number available in existing tabulations. There has been no attempt to make it uniform throughout the Handbook, which would have been a costly and laborious undertaking. In most tables at least five significant figures have been provided, and the tabular’ intervals have generally been chosen to ensure that linear interpolation will yield. fouror five-figure accuracy, which suffices in most physical applications. Users requiring higher 1 The most recent, the sixth, with F. Loesch added as cc-author, was published in 1960 by McGraw-Hill, U.S.A., and Teubner, Germany. 2 The second edition, with L. J. Comrie added as co-author, was published in two volumes in 1962 by Addison-Wesley, U.S.A., and Scientific Computing Service Ltd., Great Britain.

tional importance. Many numerical examples are given to illustrate the use of the tables and also the computation of function values which lie outside their range. At the end of the text in each chapter there is a short bibliography giving books and papers in which proofs of the mathematical properties stated in the chapter may be found. Also listed in the bibliographies are the more important numerical tables. Comprehensive lists of tables are given in the Index mentioned above, and current information on new tables is to be found in the National Research Council quarterly Mathematics of Computation (formerly Mathematical Tables and Other Aids to Computation). The ma.thematical notations used in this Handbook are those commonly adopted in standard texts, particularly Higher Transcendental Functions, Volumes 1-3, by A. ErdBlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi (McGrawHill, 1953-55). Some alternative notations have also been listed. The introduction of new symbols has been kept to a minimum, and an effort has been made to avoid the use of conflicting notation.

of the Tables precision in their interpolates may obtain them by use of higher-order interpolation procedures, described below. In certain tables many-figured function values are given at irregular intervals in the argument. An example is provided by Table 9.4. The purpose of these tables is to furnish “key values” for the checking of programs for automatic computers; no question of interpolation arises. The maximum end-figure error, or “tolerance” in the tables in this Handbook is 6/& of 1 unit everywhere in the case of the elementary functions, and 1 unit in the case of the higher functions except in a few cases where it has been permitted to rise to 2 units.

IX /-

.

INTRODUCTION

X

3.

Auxiliary

Functions

One of the objects of this Handbook is to provide tables or computing methods which enable the user to evaluate the tabulated functions over complete ranges of real values of their parameters. In order to achieve this object, frequent use has been made of auxiliary functions to remove the infinite part of the original functions at their singularities, and auxiliary arguments to co e with infinite ranges. An example will make t fi e procedure clear. The exponential integral of positive argument is given by

4.

775 ;:; E

: 89608 89717

4302 8737

d0 g. I

ze*El . 89823 .89927 90029

(z) 7113 7306 7888

8: 4 ix

:.90227 90129

4695 60”3

[ 1 ‘453

The numbers in the square brackets mean that the maximum error in a linear interpolate is 3X10m6, and that to interpolate to the full tabular accuracy five points must be used in Lagrange’s and Aitken’s methods. 8 A. C. Aitken On inte elation b iteration of roportional out the use of diherences, ‘Brot Edin i: urgh Math. 8 oc. 3.6676

recludes direct interThe logarithmic singularity polation near x=0. The Punctions Ei(x)-In x and x-liEi(ln x-r], however, are wellbehaved and readily interpolable in this region. Either will do as an auxiliary function; the latter was in fact selected as it yields slightly higher accuracy when Ei(x) is recovered. The function x-‘[Ei(x)-ln x-r] has been tabulated to nine decimals for the range 05x (&

Inequality

for

I?(~)+9(z)I~~s)llp~(~b

a

a

lJ~(z)l%iz)l’p +(Jb

“)

a

Iscd lqp

equality holds if and only if g(z) =cf(x) stant>O).

Sums

(c=con-

fj>l 3.3. Rules

for

Differentiation

and

Integration

Derivatives

equality holds if and only if jbkl=cIuEIP-’ stant>O). If p=q=2 we get Cauchy’s

(c=con-

$; (cu) =c $9 c constant

3.3.1

3.3.2

Inequality

-g (u+v)++g

3.2.9 [&

akb,]21,

q>l

.g u(v) =g

3.3.5

gi

3.2.10

& (u”) =uD e %+1.n U 2)

3.3.6 equality holds if and only (c=constant>O). If p=p=2 we get Schwa&s

3.2.11

Inequality

if

jgCs)I=clflr)Ip-’

Leibniz’s

Theorem

for

Differentiation

of an Integral

3.3.7 b(c)

d &

s

a(c)

f

(2,

wx b(c)

3

a(cj

s

b ,J(x,c)dx+f@,

4

$--f

(a,

4

2

12

ELEMENTARY

Leibniz’s

Theorem

for

Differentiation

ANALYTICAL

The following

of a Product

3.3.8 s

METHODS

formulas

S

P(x)dx

n>l

is an integer.

where P(x)

(ux”+ fJx+cy

(t&g

u+(y)

g

g

+(g

ds

g

are useful for evaluating is a polynomial

and

3.3.16 +...+~)d~rg+...+cg

dx 2 (ax2+br+c)=(4c&c-bz)~

S dX -&=1g

3.3.9

(b2--4uc

Z/COS:I z z

4.5.16

cosh2 z-sinh2

4.5.17

tanh2 z+sech2 z= 1

4.5.18

coth2 z-csch2 z= 1

Negative

Angle

z=l

Formulas

4.5.21

sinh (-.z)=-sinh

4.5.22

cash ( - z) = cash z

4.5.23

tanh (-z)=-tanh

z

Addition

4.5.24

z

z

Formulas

sinh (zl+z2) =sinh

z1 cash 22 +cosh z1 sinh 22

4.5.25

cash (zl+ 22)=cosh ZI cash

22

+sinh 4.5.26

tanh (q+zQ=(tanh

zI+tanh

z1 sinh zz &)/

(1 + tanh z1 tanh 22) FIGURE

4.6.

4.5.27

Hyperbolic jusctions.

coth (zl+zz)=(coth

zl coth %+l)/ (coth zz+coth

Relation

to Circular

Hyperbolic trigonometric

Functions

(see 4.3.49

to 4.3.34)

can be derived from formulas identities by replacing z by iz

Half-Angle

4.5.28 sinh E=(,,sh;-l)i

4.5.7

sinh z=--i

sin i;r

Formulas

ZI)

ELEMENTARY

84

TRANSCENDENTAL

FUNCTIONS

4.5.44 cash z,-cash

z2=2 s:inh (p)

sinh (v)

4.5.45 zl+tanh

sinh (21+z2) %=E;h L 2 cosh 2 1 2

coth z,+coth

sinh (2, +2,) zz=- smh 2, sinh zz

tad 4.5.46 Multiple-Angle

Formulas

4.5.31

sinh 22=2 sinh z cash ~=~~‘a~~f,

4.5.32

cash 22=2

cosh2 z--l=2

Relations

Between

Squares of COosines

4.5.47 sinh2 zL-sinh2 zz=sinh =cosh2 4.5.48 sinh2 zl+cosh2 z2=cosh =cosh2

sinh2 z-j-1

= cosh2 z + sinh2 z

Hyperbolic

Sines

(zl+zz) sinh (zl--q-cosh2 zz (q+zJ cash zl+sinh2 z2

z2)

(2,-z2)

4.5.33

tanh 2~=12+tt&annhhez,

4.5.34

sinh 32=3

4.5.35

cash 32=-3

4.5.36

sinh 42=4 sinh3 z cash 2+4 cosh3 z sinh z

4.5.49

sinh z=sinh

x cos y+i

cash x sin y

4.5.37

cash 4z=cosh4 z-j-6 sinh2 z cosh2 z+sinh’

4.5.50

cash z=cosh x cos yfi

sinh x sin y

Products

4.5.38

Hyperbolic

sinh 2+4 sinh3 z

of Hyperbolic

Sines

and

(Q-.zz>

z

and

Subtraction

of Two

De Moivre’s

(q-q)

(~~-2~)

4.5.55 z2=2 sinh (‘9)

cash (‘9)

4.5.42 sinh z,-sinh

4.5.54

4.5.56

4.5.57 z2=2 cash (9)

(cash z+sinh and

Phase

Theorem

z)“=cosh

nz+sinh

(Argument) Functions

of

m

Hyperbolic

lsinh z/ = (sinh2 x+sin2 y)* =[+(cosh 2x-cos 2y)]”

Functions

4.5.41 sinh z,+sinh

4.5.53 Modulus

(zI+zz)

Hyperbolic

Imaginary

4 . 5 .52 coth 2=sinh 2x---i sin 2y cash 2x-cos 2y

2 cash 21 cash zz=cosh (zl+zJ

+sinh Addition

and

4 . 5 .51 tanh 2=sinh 2x-j-i sin 2y cash 2x+cos 2y

Cosines

2 sinh z1 sinh zz=cosh (zl+z2)

2 sinh z1 cash z2=sinh

of Real

(z:=zfiy)

+cosh 4.5.40

in z;;;s

cash 2+4 cosh3 z

-cash 4.5.39

Functions

and

sinh (y) 4.5.58

arg sinh z=a.rctan

(coth x tan y)

(cash zl= (sinh2 x+cos2 y)” =[+(cosh 2x+cos 2y)]h arg cash z=arctan

(tanh x tan y)

cash 2x-cos 2y * ltanh Z1=(cosh 2x+cos 2y

4.5.43 cash zl+cosh z2=2 cash (‘+)

cash (y)

4.5.59

arg tanh z=arctan

~ (f:2,“,) Y

ELEMENTARY

Relations

4.5.60

TRANSCENDENTAL

Bet ween Hyperbolic

sinh x=a

CO~ll x=u

85

FUNCTIONS

(or Inverse Hyperbolic)

tanh z=a

Functions

csch x=u

sech x=,z

= --

coth x=a

sinh x----?.

a

(a’-. l)+

a(l-u2)-1

CL-’

a-‘(1-a2y

(a’- 1) -1

cash x----o

(l+u*)+

a

(l-a”)+

a-‘(l+a2)t

a-l

a(a2- 1)-i

tanh x-----

a(l+a’)-4

u-‘(a2- 1)’

a

(l+ay-+

(l-a’)+

a-’

csch x-----

am1

(a’- 1)-’

a-‘(l-a2)+

a

a(l-a2)+

(a2- 1)’

soch x-----

(l+u’)-+

a-*

(l-4)+

a(l+a2)-*

a

a-‘(a2- 1)’

coth x-----

a-‘(a2+l)f

u(a2-- 1)-i

a-l

u+a2y

(1-a2)-4

a

-

Illustration:

If sinh x=a, coth x=aB1(a2+ arcsech a=arccoth

Special Values of the Hypl>rbolic

4.5.61

Functions z

* . -a 2

0

2

1)’

(l--a2)-)

4.5.66 sech z=l

-f+&

24-go

zs+

. . . +a*

22*+

(

0

i

0

--i

co

4.5.67

z-------

1

0

-1

0

cu

coth

tanh

z------

0

-i

0

-coi

1

csch

2 _______

0~

4

OD

i

0

sech

2 _______

1

OD

-1

co

0

coth

2 _______

0~

0

co

0

1

sinh

2 ____

cash

_ __

. . .

co

2=;+;-$+$f-

IM

. . .

(l4

S S

z dt -

0 1-P

sinhm+ z cash” z dz

=-

m+n-2 + n-l

iz (t2fl)t

4.6.3 real axis from - ~0 to - 1 and + 1 to +a

sinhmel z coshn+l z

m-l -m+-

(z=x+iy)

4.6.2 real axis fro.m - 03 to + 1

+- S m+n

Oz(lTt2)1

4.6.1 imaginary axis from --im to iw

zn-l sinh z dz

m+n

S S S

The paths of integration must not cross the following cuts.

cash z dz

z-n S

sinh

n-l

4.5.86

Hyperbolic

4.6.1

4.5.85

sinh”

z dz

Definitions

csch zdz=ln

4.5.81

J

coth” z de= -%!@.?+SCoth”-2 n-l

cash z

S S S

4.5.80

Wl)

Wl>

s

4.5.79

z dz

s

z

s

4.5.78

tanh”-‘z+Stanh”-” n-l

4.5.88

Formulas

sinh z dz=cosh

4.5.77

z dz=--

z

3 -i arcsinh

z

arccsch

z

--CD

4-I

X

t-

X

-I

0 arccosh

.z

1 arcsech

z

iy I0 +I

arctanh

(m#l> 1

sinh”-’ dz z cosh”-2

5 cash”-’ z

Wl>

z FIGURE

4.7.

Branch cuts for junctions.

1 arccothz

inverse

hyperbolic

z

ELESMENTARY

arctanh z=arccoth

4.6.7

TRANSCENDENTAL

z& $ri

(see 4.5.60) Fundamental

arccoth x=i

4.6.25

(according

87

FUNCTIONS

as J&SO)

x+1 In x-l

(x2>1)

Property

The general solutions of the equations z=sinh t z=cosh t z=tanh

t

are respectively 4.6.8

t=Arcsinh

~=(-l)~

arcsinh z+kri

4.6.9

t=Arccosh

2= harccosh

4.6.10

t=Arctanh

z=arctanh

z+2k& z.+kti (k, integer)

Functions

of Negative

Arguments

FIGURE

arcsinh (- z) = - arcsinh z

4.6.11

arccosh (- z) = ai- arccosh z

*4.6.12

Addition

4.6.13 Relation

Inverse hyperbolic junctiolas.

4.8.

arctanh to

Inverse

(-2) = -arctanh Circular

and

Subtraction

z

Functions

(see

of

Two

Inverse

Hyperbolic

[2,(1

+z:)*d~

~~(1

+z;>+]

[ (zi-

1) (z$-

l)]‘}

Functions 4.4.20

to

4.6.26

4.4.25)

Hyperbolic identities can be derived from trigonometric identities by replacing 2 by i2. 4.6.14

Arcsinh

4.6.15

Arccosh z= k i Arccos

4.6.16

Arctanh

Arcsin iz

2=-i

2=

4.6.17

Arccsch z=i

4.6.18

Arcsech

4.6.19

Arccoth z=i

-i

Arcsinh

z1

&Arcsinh

z2

=Arcsinh 4.6.27

2

Arccosh

2,

Arctrm iz

f Arccosh

z2

=Arccosh

{ 2,z2k

Arccsc iz 4.6.28

2=

5 i Arcwc z

Arctanh Logarithmic

z,&Arctanh

z2=Arctanh

Arccot iz 4.6.29

Representrltions

4.6.20

arcsinh z=ln

[x+(x*-+l)t]

4.6.21

arccosh x=ln

[z+ (x2-- l)i]

4.6.22

arctanh x=4 In -1+x l-x

4.6.23

arccsch x=ln [:+($-I-1)1]

4.6.24

arcsech s=ln

[i+($-

Arcsinh z,fArccosh (x21)

=Arcsinh

/[za(l

+z:)‘&

2,(2[--

4.6.30

Arctanh

(x+0) (O ( 2ri:i;)

l)‘]

ELEMENTARY

88 Series

TRANSCENDENTAL

Expansions

FUNCTIONS

4.6.42

4.6.31 arcsinh z=z --.A2 . 3 z3++-&

-$ arccoth. 2=(1-S)+

25

Integration

1.3.5 -2.4.6.7

“+

’ ’ *

4.6.44

(lzl1)

s

(1211) 4.6.32

(l-9)

pa+1

arctanh

2

dz:=-

n+l

arctanh z -&Ssdz (n#-

1)

4.6.55 4.6.39

2 arctanhz=(l-,9)-I

4.6.40

z arccsch z=T

d

Sz arccsch ’ z(l+S)) (according as 9?z=O)

4.6.41 *See

pageII.

1 2 arcsech z= 7 dz z(l-.zy

z de=


13+i

-arctan

( =1.282475-G(.982794).

Example

Compute (.227).6gto 7D. Using 4.2.7 and Tables 4.2 and 4.4,

=e--l.o~la

ln lO)=exp

[(n+d) In lo]

=lO” exp (d In 10) 3 2>

From Table

4.4

es48=exp (E

In 10) ==exp (281.42282 42 In 10)

=10281exp (.42282 42 In 10)=lOzsl exp .97358 87

8.

(.227).8g=e. 68

(&

=exp (In 10”) exp (d In 10)

=1.282475-i(2.158799) In (2-33i)=iln

and d=the decimal part of Go*

In C.227) ,e.68(-1.48%0 b@l=

.35g46

=10281(2.647428)=(!281)2.647428. Example

12.

5262)

60.

Compute em2for x= .75 using the expansion in Chebyshev polynomials.

. ELE:MENTARY

Following 4.2.48

the procedure

TRANSCENDENTAL

in [4.3] we have from

Example

15.

Compute sin x and cos x for z=2.317 From 4.344 and Table 4.6

e-z=&m(Z) I *

91

FUNCTIONS

where a(r) are the Chebyshev polynomials defined in chapter 22. Assuming be=kO we F;WE&IZ ba, k=7,6,5,. . . 0 from the recurrence

sin (2.317)=sin

(r-2.317)=sin

(.82459 2654) =.73427

cos (2.317)=cos

(r-2.317)=

-cos

be

Linear interpolation error of 9X lo-*.

- .ooooo 00 15

6

.ooooo 0400 - .OOOOO9560

5 4

Example

.03550 4993 - .27432 74.49 .33520 2828

Example

7449)

32925 19943 !Z9577 r 08882 08665 ‘72159 62 r 48481 36811 139535 9936 r 38O=. 66322 51 r 42’=.01221 73 r

32”=.00015 51 r -38’42’32”= .675598 r. 14.

Express X= 1.6789 radians in degrees, minutes and seconds to the nearest tenth of a second. From Table 1.1 giving the mathematical constants we have 1 r-l8oo --=57.29577 r 1.6789 r=96.19388’ .19388oX6O=11.633’ .633’X60=38.0” 1.6789 r=96°11‘38.0”. *See page II.

142.

951300 . . .

17.

Compute sin x to 19D for x=.86725 13489 24685 12693. Let

in radians to 6D.

Therefore

Example

12 sin 367

The method of reduction to an angle in the first quadrant which was given in Example 15 may also be used.

13.

lo=.01745 1’=.00029 l”=.OOOOO

12 cos .867+cos

=.29612

e-.‘6=.33520 2828-(.5)(--.27432 = .47236 6553.

Express 38’42’32”

2654 gives an

16.

sin (12.867)=sin

sincef(z)=b,,-((2x-l)bl,

Example

for x=.82459

Compute sin x for x=12.867 to 8D. From 4.3.16 and Tables 4.6 and 4.8

.00018 9959 -.00300 9164

3 2 1 0

12

(.82459 2654) =-. 67885 60.

b,=(4s-2)b,+l-bc+~+AI k 7

to 71).

Table

a=.867,

/3=x-a.

From

4.3.16

and

4.6

sin (a+@)=sin a co9 @+cos a sin 0 sin a=.76239 10208 07866 22598 cos a=.64711

66288 94312 75010

With the series expansions for sin B and cos fl we compute successively 1.00000 d -2=-

00000 315

*

8’ a=

00000 OOOOO 88140 97019 16630

*

co9 /3=

.99999 8=

11859

196li

.00025 13489 24685 12693

Ba T!=-

*

!c 51-

*

sin /3=

99684

BOO25 sin a cos /3=.76239 ma a sin fl=.OOOlS sin x=.76255

2646 54842 1 13489 09967 26520 36487

22038 25351 67105 92457

57352 31308 82436 13’74

.

92

ELEMENTARY

This procedure is equivalent Taylor’s formula 3.6.4.

to interpolation

Example

TRANSCENDENTAL

with

ABC,

a=123,

B=29”16’,

cos B=(123)2+(321)2

bs=az+d--2ac

-2(123)(321) b=221.99934

cos 29’16’

00

sin A-a--= sin B b

(123)(-48887 50196)=.27086 221.99934 00

3gg18

ABC,

c2+b2-a2=81+49-16 2bc

sin A=.42591 sin B=7(*425g1

l/20= 1.52083 7931

arccot 20=;-arctan

2O=arctan

a=4,

b=7,

114

2-7 -9

===.90476

c=9,

Express z=3+9i

5954=25’12’31.6”

770g’,

in polar form.

k is an

f+2rk,

~=(3~+9~)f=

9/3=arctan

4

770g),

For k=O,

3= 1.24904 58.

exp (1.24964 58i).

24.

Compute arctan x for. z= l/3 to l2D. From 4.4.34 and 4.4.42 we have Bz.84106

8670

arctan s=arctan

(x0+@

=48Y1’22.9”

sin C=g(‘425g1

integer.

qt%=9.486833

Thus 3+93=9.486833 Example

7709

4

8396.

where r==($+y2)+,

B=arctan 1905

.05=.04995

23.

0=arctan

19.

A=.43997

arctan 20=;-arctan

z=x+iy=refe,

In the plane triangle find A, B, and C. cos A=

22.

Example

A=15’42’56.469”. Example

Example

Compute arctan 20 and arccot 20 to 9D. * Using 4.4.5,4.4.8, and Table 4.14

18.

In the plane triangle c=321; find A, b.

FUNCTIONS

Ccl.86054

h

=arctan

xo+arctan

: .L+xoh+g

=arcm

xo+(i-$&&&+~+xJ+.

803 =106’36’5.6”

.. .

where the supplementary an le must be chosen for (7. As a check we get A+ 27+C=180”00’.1”.

We have

Example

x= f = .33333 33333 33 so that h= .00033 33333 33

20.

Compute cot x for x=.4589 to 6D. Since 2.95, using Table 4.14 with interpolation in the auxiliary functionf(z) we find arcsin 2=:-[2(1-x)I?f(x)

=.32175

00000 09

05543 97.

If x is given in the form b/a it is convenient use 4.4.34 in the form b arctan ,=arctan

xO+arctan b-axe -* a+b

arcsin (.99511)=;-[2(.00489)]*j(.99511) In the present example we get =1.57079

6327-(.09889

=1.47186

2100.

388252)

(1.00040 7951)

.333

h 1 +x&+x;=

arctan jj=arctan *see page II.

.333+arctan

1 -. 3333

to

ELEMENTARY

Example

TRANSCENDENTAL

arctanh .96035=+ In 1+ .96035=$ln- 1.96035 1- .96035 .03965

25.

Compute arcsec 2.8 to 5D. Using 4.3.45 and Table 4.14

=a In 49.44136 191 =$(3.90078 7359) = 1.950394.

(z”-- I)+ arcsec z=arcsin --y .Example 27.

. [(2.8)2-l]+ arcsec 2.8=arcsm 2.8 =arcsin

93

FUNCTIONS

Compute arccosh x for x=1.5368 Using Table 4.17

.93404 97735

to 6D.

arccosh x=arccosh 1.5368=.852346 (x2-- 1)’ [(1.5368)2-l]*

= 1.20559 or using 4.3.45 and Table 4.14

arccosh 1.5368=(.852346)(1.361754)* arcsec z=arctan arcsec 2.8=arctan

(z2-l)*

=(.852346)(1.166942)

2.61533 9366

= .994638. =z-arctan

.38235 951564, f:rom 4.4.3 and 4.4.8

=1.570796-.365207

Example 28. Compute arccosh x for x=31.2 to 5D. Using Tables 4.2 and 4.17 with l/x=1/31.2 = .03205 128205

= 1.20559. Example 26.

arccosh 31.2-m

Compute arctanh 2 for x=.96035 to 6D. From 4.6.22 and Table 4.2

31.2=.692886

arccosh 31.2=.692886+3.440418=4.13330.

References Texts

14.11 B. Carlson! M. Goldstein, R,ttional approximation of functions, Los Alamos Scientific Laboratory LA-1943 (Los Alamos, N. Mex., 1955). [4.2] C. W. Clenshaw, Polynomial approximations to elementary functions, Mat!l. Tables Aids Comp. 8, 143-147 (1954). of [4.3] C. W. Clenshaw, A note c’n the summation Chebyshev series, Math. ‘l’ables Aids Comp. 9, 118-120 (1955). 14.41 G. H. Hardy, A course of pure mathematics, 9th ed. (Cambridge Univ. Press, Cambridge, England, and The Macmillan Co., New York, N.Y., 1947). [4.5] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. F’ress, Princeton, N.J., 1955). [4.6] C. Hastings, Jr., Note 143, Math. Tables Aids Comp. 6, 68 (1953). [4.7] E. W. Hobson, A treatise on plane tri~metry, 4th ed. (Cambridge Univ. Press, ambndge, England, 1918). [4.8] H. S. Wall, Analytic theory of continued fractions (D. Van Nostrand Co., Iuc., New York, N.Y., 1948). Tables

[4.9] E. P. Adams, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The l$~$onian Institution, Washington, D.C., [4.10] H.

Andoyer, Nouvelles tables tri onometrlques fondamentales (Hermann et fils, !baris, France, 1916).

[4.11] British Association for the Advancement of Science, Mathematical Tables, vol. I. Circular and hyperbolic functions, exponential, sine and cosine integrals, factorial function and allied functions, Hermitian probabilit functions, 3d ed. (Cambridge Univ. Press, 8 ambridge, England, 1951). [4.12] Chemical Rubber Corn any! Standard mathematical tables, 12th ed. ( B hemmal Rubber Publ. Co., Cleveland! Ohio, 1959). [4.13] L. J. Comne, Chambers’ six-figure mathematical tables, vol. 2 (W. R. Chambers, Ltd., London, [4.14] [4.15] Laboratory., Tables of the [4.16] Harvard. Computation function arcsin z (Harvard Univ. Press, Camz=x+iy, O