The Theory Of Functions Of A Complex Variable

344 Pages·1982·16.74 MB·English

by  A. G. Sveshnikov

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A. G. SVESHNOKOV. A. N. TIKHONOV
MIR PUBLISHERS

A. r. CBEIDHHHOB, A. H. THXOHOB
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H3.ll:ATEJlbCTBO «HAYHA>>
A. G. SVESHNIKOV and A. N. TIKHONOV
THE THEORY
OF FUNCTIONS
OF A COMPLEX
VARIABLE
Translated from the Russian
by GEORGE YANKOVSKY
MIR PUBLISHERS · MOSCOW
First published 1971
Second printing 1973
Second edition 1978
Second edition, second printing 1982
@ liaAaTeJILCTBO cHayKU, 1974, C IIBMeReRHRMH
© English translation, Mir Publishers, 1978
CONTENTS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Chapter 1. THE COMPLEX VARIABLE AND FUNCTIONS OF A
COMPLEX VARIABLE 11
1.1. Complex Numbers and Operations on Complex Numbers H
a. The concept of a complex number . . . . . . . . 11
b. Operations on complex numbers . . . . . . . . . 11
c. The geometric interpretation of complex numbers . 13
d. Extracting the root of a complex number 15
1.2. The Limit of a Sequence of Complex Numbers 17
a. The definition of a convergent sequence . 17
b. Cauchy's test . . . . . . . . . . . . • 19
c. Point at infinity • . . . . . . . • . . 1!}
1.3. The Concept of a Function of a Complex Variable. Continuity 20
a. Basic definitions 20
b. Continuity . . . . . . . . . . . . . . . 23
c. Examples . . . . • . . . . . . . . . . . 26
1.~. Ditlerentiating the Function of a Complex Variable 30
a. Definition. Cauchy-Riemann conditions •.. 30
b. Properties of analytic functions . . . . . . . . . . . 33
c. The geometric meaning of the derivative of a function of a
complex variable . . . . . . . . . . . 35
d. Examples . . . . . . . . . . . . . . 37
·1.5. An Integral with Respect to a Complex Variable 38
a. Basic properties . 38
b. Cauchy's Theorem 41
c. Indefinite integral . 44
1.6. Cauchy's Integral . . 47
a. Deriving Cauchy's formula .. 47
b. Corollaries to Cauchy's formula . . . . . . . . . . . 50
c. The maximum-modulus principle of an analytic function 51
t . 7. Integrals Dependent on a Parameter . • . . . . . . . . . 53
a. Analytic dependence on a parameter . . . . . . . . . . 53
b. An analytic function and the existence of derivatives of all
orders • . . . • • . . • • • • • . 55
Chapter 2. SERIES OF ANALYTIC FUNCTIONS • 58
2.1. Uniformly Convergent Series of Functions of a Complex
Variable .........•............. 58
6 Contents
a. Number series . . . . . . . . . . . . . 58
b. Functional series. Uniform convergence . . 59
c. Properties of uniformly convergent series. Weierstrass'
theorems . . . . . . . . . . 62
d. Improper integrals dependent on a parameter 66
2.2. Power Series. Taylor's Series 67
a. Abel's theorem .. 67
b. Taylor's series . . 72
c. Examples 74
2.3. Uniqueness of Definition of an Anal)1ic Function 76
a. Zeros of an analytic function 76
b. Uniqueness theorem . . . . . . . . . • • . 77
Chapter 3. ANALYTIC CONTINUATION. ELEMENTARY
FUNCTIONS OF A COMPLEX VARIABLE 80
3.t. Elementary Functions of a Complex Variable. Continuation
from the Real Axis . . . . . . . . . . . • . . . . . . 80
a. Continuation from the real axis . 80
b. Continuation of relations . . . . 84
c. Properties of elementary functions 87
d. Mappings of elementary functions 91
3.2. Analytic Continuation. The Riemann Surface • 95
a. Basic principles. The concept of a Riemann surface 95
b. Analytic continuation across a boundary . . . . . . . . 98
c. Examples in constructing analytic continuations. Con-
tinuation across a boundary . . . . . . . . . . . . . 100
d. Examples in constructing analytic continuations. Con-
tinuation by means of power series . . . . . . . . . . 105
e. Regular and singular points of an analytic function . . . 108
f. The concept of a complete analytic function . . . • . . 111
Chapter 4. THE LAURENT SERIES AND ISOLATED SINGULAR
POINTS . . . . . . . . . . . • . . . . . • 113
4.1. The Laurent Series . . . . . . . . . • . . . . . • • 113
a. The domain of convergence of a Laurent series . . . 113
h. Expansion of an analytic function in a Laurent series t 15
4.2. A Classification of the Isolated Singular Points of a Single-
Valued Analytic Function . . . . . . . . 118
Chapter 5. RESIDUES AND THEIR APPLICATIONS . • • • . • • 125
5.t. The Residue of an Analytic Function at an Isolated Singu-
larity . . . . . . . . . . . . . . . . . . . .. • . . . 125
a. Definition of a residue. Formulas for evaluating residues 125
b. The residue theorem . . . . . . . . . . . . 127
5.2. Evaluation of Definite Integrals by !\leans of Residues 130
· 2n
a. Integrals of the form ) R (cos e, sin e) de 131
0