AUTHOR'S PREFACE SECOND FRENCH EDITION
The first part of this volume has undergone only slight changes,
while the rather important modifications that have been made
appear only in the last chapters.
In the first edition I was able to devote but a few pages to par-
tial differential equations of the second order and to the calculus
of variations. In order to present in a less summary manner such
broad subjects, I have concluded to defer them to a third volume,
which will contain also a sketch of the recent theory of integral
equations. The suppression of the last chapter has enabled me to
make some additions, of which the most important relate to linear
differential equations and to partial differential equations of the
firSt Order " E. GOUESAT
iii
TRANSLATORS' PREFACE
As the title indicates, the present volume is a translation of the
first half of the second volume of Goursat's "Cours d' Analyse." The
decision to publish the translation in two parts is due to the evi-
dent adaptation of these two portions to the introductory courses in
American colleges and universities in the theory of functions and
in differential equations, respectively.
After the cordial reception given to the translation of Goursat's
first volume, the continuation was assured. That it has been
delayed so long was due, in the first instance, to our desire to await
the appearance of the second edition of the second volume in
French. The advantage in doing so will be obvious to those who
have observed the radical changes made in the second (French)
edition of the second volume. Volume I was not altered so radi-
cally, so that the present English translation of that volume may be
used conveniently as a companion to this ; but references are given
here to both editions pf the first volume, to avoid any possible
difficulty in this connection.
Our thanks are due to Professor Goursat, who has kindly given
us his permission to make this translation, and has approved of the
plan of publication in two parts. He has also seen all proofs in
English and has approved a few minor alterations made in transla-
tion as well as the translators' notes. The responsibility for the
latter rests, however, with the translators.
E. R. HEDRICK
OTTO DUNKEL
Author(s): Édouard Goursat, Earle Raymond Hedrick and Otto Dunkel
Edition: 1
Publisher: Ginn & company
Year: 1916
Language: English
Pages: 271
City: Boston, New York
Tags: Calculus, Complex Variables
CONTENTS
PAGE
CHAPTER I. ELEMENTS OF THE THEORY 3
I. GENERAL PRINCIPLES. ANALYTIC FUNCTIONS 3
1. Definitions 3
2. Continuous functions of a complex variable 6
3. Analytic functions 7
4. Functions analytic throughout a region 11
5. Rational functions 12
6. Certain irrational functions 13
7. Single-valued and multiple-valued functions 17
II. POWER SERIES WITH COMPLEX TERMS. ELEMENTARY
TRANSCENDENTAL FUNCTIONS 18
8. Circle of convergence 18
9. Double series 21
10. Development of an infinite product in power series .... 22
11. The exponential function 23
12. Trigonometric functions 26
13. Logarithms 28
14. Inverse functions : arc sin z, arc tan z 30
15. Application to the integral calculus 33
16. Decomposition of a rational function of sin z and cos z into
simple elements 35
17. Expansion of Log (1 + z) 38
18. Extension of the binomial formula 40
III. CONFORMAL REPRESENTATION 42
19. Geometric interpretation of the derivative 42
20. Conformal transformations in general 45
21. Conformal representation of one plane on another plane . . 48
22. Rlemann's theorem 50
23. Geographic maps 52
24. Isothermal curves 54
EXERCISES 56
vii
viii CONTENTS
PAGE
CHAPTER II. THE GENERAL THEORY OF ANALYTIC FUNC-
TIONS ACCORDING TO CAUCHY 60
i
I. DEFINITE INTEGRALS TAKEN BETWEEN IMAGINARY LIMITS 60
25. Definitions and general principles 60
26. Change of variables . 62
27. The formulae of Weierstrass and Darboux 64
28. Integrals taken along a closed curve 66
31. Generalization of the formulae of the integral calculus . . 72
32. Another proof of the preceding results 74
II. CAUCHY'S INTEGRAL. TAYLOR'S AND LAURENT'S SERIES.
SINGULAR POINTS. RESIDUES . 75
33. The fundamental formula 75
34. Morera's theorem 78
35. Taylor's series 78
36. Liouville's theorem 81
"37.. Laurent's series 81
38. Other series 84
39. Series of analytic functions 86
40. Poles 88
41. Functions analytic except for poles 90
42. Essentially singular points 91
43. Residues 94
III. APPLICATIONS OF THE GENERAL THEOREMS 95
44. Introductory remarks 95
45. Evaluation of elementary definite integrals 96
46. Various definite integrals 97
47. Evaluation of FQo) T(l-p) 100
48. Application to functions analytic except for poles .... 101
49. Application to the theory of equations 103
50. Jensen's formula 104
51. Lagrange's formula 106
52. Study of functions for infinite values of the variable . . . 109
IV. PERIODS OF DEFINITE INTEGRALS . 112
53. Polar periods _. ._ 112
54. A study of the integral J^rfc/Vl z 2 . 114
55. Periods of hyperelliptic integrals 116
56. Periods of elliptic integrals of the first kind 120
EXERCISES 122
CONTENTS ix
PAGE
CHAPTER III. SINGLE-VALUED ANALYTIC FUNCTIONS . . 127
I. WEIERSTRASS'S PRIMARY FUNCTIONS. MITTAG-LEFFLER'S
THEOREM 127
57. Expression of an integral function as a product of primary
functions 127
58. The class of an integral function 132
59. Single-valued analytic functions with a finite number of
singular points 132
60. Single-valued analytic functions with an infinite number of
singular points 134
61. Mittag-Leffler's theorem , . . 134
62. Certain special cases 137
63. Cauchy's method .139
64. Expansion of ctn x and of sin x 142
II. DOUBLY PERIODIC FUNCTIONS. ELLIPTIC FUNCTIONS . 145
65. Periodic functions. Expansion in series 145
66. Impossibility of a single-valued analytic function with
three periods 147
67. Doubly periodic functions 149
68. Elliptic functions. General properties . 150
69. The function p (w) 154
70. The algebraic relation between p (M) and p' (w) 158
71. The function f (M) 159
72. The function
73. General expressions for elliptic functions 163
74. Addition formulae 166
75. Integration of elliptic functions 168
76. The function 170
III. INVERSE FUNCTIONS. CURVES OF DEFICIENCY ONE . . 172
77. Relations between the periods and the invariants .... 172
78. The inverse function to the elliptic integral of the first kind 174
79. A new definition of p (w) by means of the invariants . . . 182
80. Application to cubics in a plane 184
81. General formulae for parameter representation 187
82. Curves of deficiency one 191
EXERCISES 193
CHAPTER IV. ANALYTIC EXTENSION 196
I. DEFINITION OF AN ANALYTIC FUNCTION BY MEANS OF
ONE OF ITS ELEMENTS 196
83. Introduction to analytic extension 196
84. New definition of analytic functions 199
x CONTENTS
PAGE
85. Singular points 204
86. General problem 206
II. NATURAL BOUNDARIES. CUTS 208
87. Singular lines. Natural boundaries 208
88. Examples 211
89. Singularities of analytical expressions 213
90. Hermite's formula 215
EXERCISES . 217
CHAPTER V. ANALYTIC FUNCTIONS OF SEVERAL VARIABLES 219
I. GENERAL PROPERTIES 219
91. Definitions 219
92. Associated circles of convergence 220
93. Double integrals 222
94. Extension of Cauchy's theorems 225
95. Functions represented by definite integrals 227
96. Application to the F function 229
97. Analytic extension of a function of two variables . . . 231
II. IMPLICIT FUNCTIONS. ALGEBRAIC FUNCTIONS .... 232
98. Weierstrass's theorem 232
99. Critical points 236
100. Algebraic functions 240
101. Abelian integrals ... 243
102. Abel's theorem 244
103. Application to hyperelliptic integrals 247
104. Extension of Lagrange's formula 250
EXERCISES 252
INDEX 253
