A Treatise on the Integral Calculus with Applications, Examples and Problems [II, Reprint 1954 ed.]

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First Edition, Macmillan; 192 2 Reprinted, Chelsea, 1954

PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS. CHAPTER XXIII. CHANGE OF THE VARIABLES IN A MULTIPLE

I~"'TEGRAL. PAGES

ARTS.

1-13 13-27

826-829. 830-832. 833-840.

Change of Order of Integration Change of the Variables Jacobians. BERTRAND'S Definition and Method. of Calculation

27-33

841-852.

The General Multiple I ntegral PROBLEMS -

33-43 44-48

CHAPTER XXIV. EULERIAN INTEGRALS, GAUSS' II FUNCI'ION. 853-870.

Beta and Gamma Functions

871-874.

n-1 I,t" x-1- dx; Y(n)r(l-n)='lr/sinmr; Gauss' Theorem, .0 +x

-

a Case

49-61

61-65

875-885. 886-896. 897-909. 910-930.

Theorems of WALLIS and STIRLING The II Function EULER'SConstant. GAUSS' Theorem Expansions of log F(x+ 1), etc. Calculation of F Functions. Definite Integrals .

65-76 76·86 87-96

931-936.

Approximate Summation CAUCHY'S Theorems; Approximation Formulae; DE MORGAN'S Investigation Integration by Continued Fractions. Factorial Series for y,(a+x); GAUSS' Calculation of "'(x) TABULATED RESULTS PROBLEMs -

116-120

937-946. 947-954. 955·957.

vii

96-116

121-130 130-139 139-144 144-152

viii

CONTENTS.

CHAPTER XXV. LEJEUNE-DIRICHLET INTEGRALS, LIOUVILLE INTEGRALS. PAGES

ARTS.

958-967. 968-984. 985-987.

DIRICHLET'S Theorem Extensions by LIOUVILLE, BOOLE, CATALAN. Generalisations LIOUVILLE'S Integral. LIOUVILLE'S Proof of GAUSS' Theorem PROBLEMS -

153-160 160-173 173-175 176-179

CHAPTER XXVI. DEFINITE INTEGRALS (I.). 7f

988-992. General Summary of Methods;

1"'" log sin ...c dx ;

Illus-

trations t" sinmrx 993-1027. Form '0 --ft-dx(m4:n), etc. .0 x 1028-1030. WOLSTENHOLME'S Principle 1031-1034. 1035-1041.

('"

'0

.0

• P Sln,q'--c

1'"

;t

d» resumed.

e-(lZ - e-n

x

o

t"

cos . b» d»,

.sina.t'

- - d»,

.r.

1042-1070. Various Forms.

r

Jo

207-208

['" .0

-

etc. 2

e-e2(a ,,2+i'! ) dx,

1'" aC~S1X2dz, . etc. ; +.1:

f'"

~nd.x . _'" (.v-b)2+ a2' etc.;

cos rx dor.

('" ./0

213-216

0

~~d.x x(a:4+2a2.1,·2cos2a+a4)' x l ft + a 2ft'

209-213

e-z • cos a» d»,

.0

('" cos rx d.x ./0

f'"

188-207

Partial Fractions·

sin

'0 e- x

•0

180-188

x 4ft

-

2a 2ft.v1ft cos 2na + a 1ft ,

PROBLEMS -

etc.

217-236 -

237-243

CHAPTER XXVII. DEFINITE INTEGRALS (II.). LOGARITHMIC AND EXPONENTIAL FUNCl'IONS INVOLVED. 1071-1076. Series required 1077·1085. Groups A, B, ... G.

f (I

~.q(

lOg1r ±x'')s d»

244-247 Various Forms of Type 248-259

ix

CONTENTS. A.RTS.

PA.GES

(ooIF(.l')~, etc,

1086-1092. General Principles.

260·263

)0 +,xfl.r:

...

1093-1096. 10"9" (log sin 0)2dO, etc.

263-265

Y'+l :.v" d»

265-266

KUMMER'S Integrals

266-269

oo

1097-1098. WOLSTENHOLME'S Expressions for 1099-1100. Group H.

LEGENDRE'S Rule.

00

1101-1103. Groups I, J. Derivations from 1104-1107. Group K.

Type

-.-1loo coshpx

.0

sin 1 qx

[

.0

. S1l1

L 0

( log! 1

,xll-l

1-- d.v (1 >a>O)

±,x

274-278

mx dx

1108-1120. Groups L, M, N. Formulae of POISSON, ABEL, CAUCHY, LEGENDRE PROBLEMS

269-274

278·286 287-292

-

CHAPTER XXVIII. DEFINITE INTEGRALS (IlL).

i" .vpsinan.r:d,x, etc.

1121-1133. Types t" cosSl.vcosqxd.t', etc.;

Jo

)0

1134-1158. Results derivable from well-known Series.

t" cospO )0 (f-=-2a cos-0+ a2)ft «o

293-301

Type

-

301-320

1159· 11M. SERRET'S Investigation of ['" e- kx.rfl - 1 d.t', k complex.

./0

Deductions

321-327

1169-1181. FRESNEL'S Integrals.

SOLDNER'S Function li(.t')

327-336

1182-1188. FRULLANI'S Theorem. ELLIOTT'S and LEUDESDOR:r'S Extensions

337-342

1189-1201. Complex Constants -

342-352

PROBLEMS

-

353-361

CHAPTER XXIX. THE COMPLEX VARIABLE. VECTORS. REPRESENTATION.

1202-1221. Vectors. Laws of Combination. DEMOIVRE'S Theorem 1222-1234. Continuity, etc.

CONFORMAL

ARGAND Diagram.

Revision of Definitions .

362-376 376·382

x

CONTENTS. PAGES

ARTS.

1235·1242. Conformal Representation. WEIERSTRASS' Example of a Function with no determinable Differential Coefficient. Differentiation of a Complex Function 1243·1246. Definitions. Zeros, Singularities, etc. 1247·1255. Conformal Representation. Isogonal Property, Connection of Elements of Area. Connection of Curvatures. Curvature Formulae. Quasi-Inversion. Homographic Relation 1256·1262. Branch Points. RIEMANN'S Surfaces . 1263-1265. Number of Roots in a given Contour PRO:BLEMS

-

382-387 388-389

389-402 402-406 407-415 415-418

CHAPTER XXX. INTEGRATION.

CONTOUR INTEGRATION. THEOREM.

TAYLOR'S

1266·1274. Definitions. General Properties. Integration of a Series 1275·1285. CAUCHY'S Theorem. Deformation of a Path. Loops. Exclusion of Singularities.

423-434

1286-1287.

434·436

f

ep(z)dz z-a'

f

ep(z)dz

(z-a 1)(z-a2 )

...

(z-a,)

1288-1297. Branch-Points and their Effect 1298. Period- Parallelograms 1299·1300. Differential Coefficients of a Synectic Function. TAYLOR'S Theorem 1301·1328. Contour Integration PROBLEMS -

419·423

436·446 446-447 448-452 452-478 479-482

CHAPTER XXXI. ELLIPTIC INTEGRALS AND FUNCTIONS. 1329-1342. Periodicity 1343. sn LU, cn LU, dn LU 1344-1354. Addition Formulae and Deductions. JACOBI'S 33 Formulae. Periodicity 1355·1361. Duplication, Dimidiation, Triplication 1362-1365. The General Proposition for Addition Formulae. Second and Third Classes of Legendrian Integrals

483-498 498 498·508 508·511 511-513

xi

CONTENTS. ARTS.

PAGBS

1366-1379. Jacobian Zeta, Eta, Theta Functions.

involving Jacobian Functions . PROBLDIS

Integrations -

514-519 520-529

-

CHAPTER XXXII. ELLIPl'IC INTEGRALS. THE WEIERSTRASSIAN FORMS. The Differential Coefficients of p(u) - 530-532 - 532-534 1385-1386. Periodicity of P(u) 1387-1413. AdditionFormulaforp(u). Deductions. p(nu) -P(u). 1380-1384. Definitions of p(u), '('II), 0'('11).

SCHWARZ -

-

534-545

1414-1415. Connection of the Weierstrassian and Legendrian

Periods and Functions

545-547

1416-1419. Expansions of the Weierstrassian Functions

. 1420-1431. Addition Formulae for Zeta and Sigma Functions and Deductions 1432-1445. Differentiation of P(u) and Integration by Aid of Weierstrassian Functions -

547-549

PROBLEMS

-

-

549-553 553-560 561-566

CHAPTER XXXIII. ELUPTIC FUNCTIONS.

REDUCTION TO STANDARD FORMS.

1446-1458. Reduction to Weierstrassian Form 1459-1479. Reduction to Legendrian Form 1480-1481.

LANDEN'S

Transformation. Illustrative Examples -

567-578 578-593 594-597

-

598-603

Notation. Variation of an Integral 1499-1502. Conditions for a Stationary Value. The Equation

604-614

PRoBLEMS

-

CHAPTER XXXIV.

(SECTION

I.)

CALCULUS OF VARIATIONS. 1482-1498. The Symbols 0, w.

K =0

-

Case of V dependent on Terminals 1504-1507. Relative Maxima and Minima. LAGRANGE'S Rule. V a Perfect Differential . 1503.

614-620 621 621-629

CONTENTS.

Xli

PAGBS

ARTS.

1508-1523. Several Dependent Variables 1524-1546. Geodesics. Least Action. Brachistochronism. Energy Condition of Equilibrium . PROBLEMS -

CHAPTER XXXIV.

629-639 639-650 650·660

II.)

(SECTION

DOUBLE INTEGRALS. CULVERWELL'S METHOD OF DISCRIMINATION. 1547-1557. Double Integrals. Two Independent Variables 1558-1564. Relative Maxima and Minima. Bubbles. Limited Variation· 1565.

Stationary Value of

f[uss; with condition

ffWd;rdy=a.'·

-

-

.

-

.

1566·1583. Discrimination. CULVERWELL'S Method. Points. Various Cases BIBLIOGRAPHY 1584. PROBLEMS -

CHAPTER XXXV.

(SECTION

661-669 669-673

673-674

Conjugate 674-691 691 692

I.)

FORMULAE OF LAGRANGE AND FOURIER. 1585-1596. FOURIER'S Theorem. The Cosine Series. The Sine Series 1597-1604. A Remarkable Limiting Form. POISSON'S Investigation 1605·1608. GRAPHICAL REPRESENTATION OF THE RESULTS 1609-1615. REMARKS AND ILLUSTRATIONS PROBLEMS -

CHAPTER XXXV.

(SECTION

693-699 699-710 710·712 712-716 717-719

II.)

DIRICHLET'S INVESTIGATION.

1616·1625. DnuCHLET'S Investigation. Existence of Maxima and Minima, Discontinuities 1626-1627. Application to FOURIER'S Series. CAUCHY'S Identity 1628·1632. The

Limit

Ltw"+w rio sin.(J).r cf>(x) d». .10 .r

Illustrations 1633-1639. Various Deductions. PROBLEMS -

FOURIER'S Formula

720-728 729·730

Graphical 731-734 734-736 737-744

xiii

CONTENTS.

CHAPTER XXXVI. MEAN VALUES. ARTS.

PAGES

1640-1650. General Conception. Combination of Means. Density

of a Distribution 1651-1654. The Inverse Distance.

745-754

Theorems on Potential.

Typical Examples

754-757

1655-1656. A Useful Artifice

757-766

Mean Areas and Volumes. Miscellaneous Means 1664-1670. Certain Inequalities, and their Consequences. General Means in Terms of Restricted Means 1671-1679. CLERK MAXWELL'S "Geometric Mean," useful for Electromagnetic Problems PROBLEMS -

1657-1663. M(p2), M(pfl).

766-778' 779-782 782-786 786·791

CHAPTER XXXVII. CHANCE. 1680-1687. Definitions and Illustrations of Applications of the

Calculus -

792-800

1688-1691. Random Points

800-802

1692·1695. Condensation Graph. Illustrations1696-1703. Inverse Probability. BAYES' Problem 1704:-1706. BUFFON'S Problem. Parallel Rulings. Rectangular

802-819 820-825

Rulings

825-827

1707-1711. Random Lines-

828-830

1712-1722. Number of Lines crossing a Contour.

Various Cases

1723-1735. Theorems of SYLVESTER and Oaozrox 1736-1743. D'ALEMBERT'S Mortality Curve.

Definitions.

830-834 834-844

Ex-

pectation of Life PROBLEMS -

844-848 849-852

CHAPTER XXXVIII. UNCERTAINTIES OF OBSERVATION. Frequency Law. Weight. Modulus Mean Error. Mean of Squares. Error of Mean Square. Probable Error .

1744-1749. LAPLACE'S Hypothesis. 1750.

853-857 857-858

xiv

CONTENTS. P.\GES

ARTS.

1751-1752. KRAMP'S Table. The Graph 1753-1756. Resultant Weight. Weight of a Series of Observations 1757-1758. Determination of Error of Mean Square from " Apparent Errors " 1759-1762. Reduction to Linear Form of Equations connecting Errors of a System of Physical Elements. Standardisation 1763. PRINCIPLE OFLEAST SQUARES 1764-1779. The Normal Equations. Order of Procedure. BmLIOGRAPHY. ILLUSTRATIONS PROBLEMS

858-859 859·862 862-863

863-865 865 866-875 876-878

CHAPTER XXXIX. THEOREMS OF STOKES AND GREEN.

HARMONIC ANALYSIS.

1780-1781. STOKEs'Theorem 1782-1784. GREEN'S Theorem. Deductions . 1785-1797. Harmonic Analysis. Spherical, Solid and Surface Harmonics. Poles and Axes 1798-1808. LEGENDRE'S Coefficients. Zonal Harmonics. Properties. Power Series. RODRIGUES' Form. Various Expressions for P (p) 1809-1812. Limiting Values. Expressions in terms of Definite Integrals 1813·1814. Various Forms of LAPLACE'S Equation 1815-1816. The Differential Equation satisfied by a LEGENDRE'S Coefficient. General Solution 1817·1826.

fl fl Pft2dp=2n~1 fl;;:

879-881 882-886 886·890

890·894 894-895 895-897 897·898

PmP n dp =0, m =1= n. P n in terms of p. Deduc-

-1

tions

1827. 1828-1831.

898-902 902

Calculation of the Coefficient of P n in p"

902·905

1832·1833. Expansion of f(p) in terms of LEGENDRE'S Coefficients, Series Unique 1834-1835. P n', P n", eto., in terms of LEGENDRE'S Coefficients

905·906

1836-1837.

dp.

fl

Pm'Pn'dp

1838·1849. IVORY'S Equation.

905

906·907

Various Working Theorems·

907-9lJ

xv

CONTENTS. ARTS.

PAGES

1850-1852. Illustrative Applications 1853.

hST OF WORKING FORMULAE.

1854:-1856. Roots of P n =0. Graphs of r=aPn , r =a(I+EPn) 1857-1860. I(2n+ I)Pn' A Discontinuity. Physical Meaning 1861-1863. Practical Expression of a Rational Function in Solid Harmonics 1864. 1865-1867. 1868-1869. 1870-1871. 1872. 1873-1874.

Change of Axis Tesseral and Sectorial Harmonics Expansion of F(jJ., ¢). O'BRIEN'S Proof Integral of Product of two Harmonics over Unit Sphere Theorem as to V within or without the Sphere when known on the Surface Differentiation of Zonal Harmonics. Change of Origin -

911-912 913 914-917 917-920 920-921 921-923 923-925 925-927 927-928 928 929-930 931-939

PROBLEMS

CHAPTER XL. SUPPLEMENTARY NOTES. 1875-1883 A. RIEMANN'S Definition and Theorem 1884-1889 B. Convergence of an Infinite Integral 1890 C. Strict Accuracy of Standard Forms -

940-945 945-948 948-949

1891-1893 D. HERMITE'S Method for Rational Fractions

949-952

1894

E. LEGENDRE'S Transformation applied to Form

JXa;y 1895-1899 F. Continuity of a Function of two Independent Variables. Differentiation of a Definite Integral. Double Integrals 1900-1902 G. Uniform Convergence. Double Limits 1903-1904:H. Unicursal Curves 1905-1910 I. GENERAL REVIEW

952-953

PROBLEMS -

953-956 956-957 957-959 959·962 962-966

ANSWERS -

966-974

IYDEX

975-980