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

Table of contents :
Title Page
Preface
Table of Contents
CHAPTER XXIII - CHANGE OF THE VARIABLES IN A MULTIPLE INTEGRAL.
CHAPTER XXIV - EULERIAN INTEGRALS, GAUSS' Pi(x,µ) FUNCTION.
CHAPTER XXV - LEJEUNE-DIRICHLET INTEGRALS, LIOUVILLE INTEGRALS.
CHAPTER XXVI - DEFINITE INTEGRALS (I.).
CHAPTER XXVII - DEFINITE INTEGRALS (II.).LOGARITHMIC AND EXPONENTIAL FUNCl'IONS INVOLVED.
CHAPTER XXVIII - DEFINITE INTEGRALS (IlI.).
CHAPTER XXIX - THE COMPLEX VARIABLE. VECTORS. CONFORMAL REPRESENTATION.
CHAPTER XXX - INTEGRATION. CONTOUR INTEGRATION. TAYLOR'S THEOREM.
CHAPTER XXXI - ELLIPTIC INTEGRALS AND FUNCTIONS.
CHAPTER XXXII - ELLIPl'IC INTEGRALS. THE WEIERSTRASSIAN FORMS.
CHAPTER XXXIII - ELLIPTIC FUNCTIONS. REDUCTION TO STANDARD FORMS.
CHAPTER XXXIV - (SECTION I.) CALCULUS OF VARIATIONS.
CHAPTER XXXIV - (SECTION II.) DOUBLE INTEGRALS. CULVERWELL'S METHOD OF DISCRIMINATION.
CHAPTER XXXV - (SECTION I.) FORMULAE OF LAGRANGE AND FOURIER.
CHAPTER XXXV - (SECTION II.) DIRICHLET'S INVESTIGATION.
CHAPTER XXXVI - MEAN VALUES.
CHAPTER XXXVII - CHANCE.
CHAPTER XXXVIII - UNCERTAINTIES OF OBSERVATION.
CHAPTER XXXIX - THEOREMS OF STOKES AND GREEN. HARMONIC ANALYSIS.
CHAPTER XL.- SUPPLEMENTARY NOTES.
A. RIEMANN'S Definition and Theorem
B. Convergence of an Infinite Integral
C. Strict Accuracy of Standard Forms
D. HERMITE'S Method for Rational Fractions
E. LEGENDRE'S Transformation applied to Form Int(1/(X*Sqrt(Y)), dx)
F. Continuity of a Function of two IndependentVariables. Differentiation of a Definite Integral. Double Integrals
G. Uniform Convergence. Double Limits
H. Unicursal Curves
I. GENERAL REVIEW
ANSWERS TO EXAMPLES AND PROBLEMS.
INDEX
Chelsea Scientic Books

Citation preview

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