A. G. SVESHNIKOV, A. N. TIKHONOV

The Theory

of Functions

of a Complex Variable

Translated from the Russian by

GEORGE YANKOVSKY

MIR PUBLISHERS • MOSCOW 1973

CONTENTS

Introduction 9

Chapter 1. THE COMPLEX VARIABLE AND FUNCTIONS OF A COM­PLEX VARIABLE 11

1. Complex Numbers and Operations on Complex Numbers . . . 11 1. The concept of a complex number 11 2. Operations on complex numbers 11 3. The geometric interpretation of complex numbers 12 4. Extracting the root of a complex number 15

2. The Limit of a Sequence of Complex Numbers 17 1. The definition of a convergent sequence 17 2. Cauchy's test 19 3. Point at infinity 19

3. The Concept of a Function of a Complex Variable. Continuity 20 1. Basic definitions 20 2. Continuity 23 3. Examples 25

4. Differentiating the Function of a Complex Variable 30 1. Definition. Cauchy-Riemann conditions 30 2. Properties of analytic functions 33 3. The geometric meaning of the derivative of a function of

a complex variable 34 4. Examples 36

5. An Integral with Respect to a Complex Variable 38 1. Basic properties 38 2. Cauchy's Theorem 40 3. Indefinite integral 43

6. Cauchy's Integral 46 1. Deriving Cauchy's formula 46 2. Corollaries to Cauchy's formula 48 3. The maximum-modulus principle of an analytic function 49

7. Integrals Dependent on a Parameter 51 1. Analytic dependence oh a parameter 51 2. An analytic function and the existence of derivatives of all

orders *. 53

Chapter 2. SERIES OF ANALYTIC FUNCTIONS 56

1. Uniformly Convergent Series of Functions of a Complex Va­riable 56

1. Number series 56 2. Functional series. Uniform convergence 57 3. Properties of uniformly convergent series. Weierstrass" theo­

rems 60

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2. Power Series. Taylor's Series 64 1. Abel's Theorem 64 2. Taylor's series 68

3. Uniqueness of Definition of an Analytic Function 72 1. Zeros of an analytic function 72 2. Uniqueness theorem , 72

Chapter 3. ANALYTIC CONTINUATION. ELEMENTARY FUNCTIONS OF A COMPLEX VARIABLE 76

1. Elementary Functions of a Complex Variable. Continuation from the Real Axis 76

1. Continuation from the real axis 76 2. Continuation of relations 80 3. Properties of elementary functions 83 4. Mappings of elementary functions 86

2. Analytic Continuation. The Riemann Surface 91 1. Basic principles. The concept of a Riemann surface . . . 91 2. Analytic continuation across a boundary 94 3. Examples in constructing analytic continuations. Continua­

tion across a boundary 95 4. Examples in constructing analytic.continuations. Continua­

tion by means of power series 100 5. Regular and singular points of an analytic lunction . . . 103 6. The concept of a complete analytic function 107

Chapter 4. THE LAURENT SERIES AND ISOLATED SINGULAR POINTS 109

1. The Laurent Series 109 1. The domain of convergence of a Laurent series 109 2. Expansion of an analytic function in a Laurent series . . Ill

2. A classification of the Isolated Singular Points of a Single-Valued Analytic Function 114

Chapter 5. RESIDUES AND THEIR APPLICATIONS 121 1. The Residue of an Analytic Function at an Isolated Singularity 121

1. Definition of a residue Formulas for evaluating residues 121 2. The residue theorem 123

2. Evaluation of Definite Integrals by Means of Residues . . . . 125 2n

1. Integrals of the torm С /?(cos6, sin9)d9 125 о

OD

2. Integrals of the form [ f{x)dx 127 - CO

CO

3. Integrals of the form \ eiaX / {x) dx. Jordan's lemma . . 130 - CO

4. The case of multiple-valued functions 136

Contents 7

8. Logarithmic Residue 142 1. The concept of a logarithmic residue 142 2. Counting the number of zeros of an analytic function . . 143

Chapter 6. CONFORMAL MAPPING 148

1. General Properties 148 1. Definition of a conformal mapping 148 2. Elementary examples 152 3. Basic principles 155 4. Riemann's theorem 160

2. Linear-Fractional Function 164 3. Zhukovsky's Function 174 4. Schwartz-Christoffel Integral. Transformation of Polygons . . . 177

Chapter 7. ANALYTIC FUNCTIONS IN THE SOLUTION OF BOUN­DARY-VALUE PROBLEMS 186

1. Generalities 186 1. The relationship of analytic and harmonic functions . . . 186 2. Preservation of the Laplace operator in a conformal mapping 187 3. Dirichlet's problem 189 4. Constructing a source function 192

2. Applications to Problems in Mechanics and Physics 194 1. Two-dimensional steady-state motion of a fluid 194 2. A two-dimensional electrostatic field 206

Chapter 8. FUNDAMENTALS OF OPERATIONAL CALCULUS 216

1. Basic Properties of the Laplace Transformation 216 1. Definition 216 2. Transforms of elementary functions 220 3. Properties of a transform 222 4. Table of properties of transforms 230 5. Table of transforms 230

2. Determining the Original Function from the T r a n s f o r m . . . . 232 1. Mellin's formula 232 2. Existence conditions of the original function 235 3. Computing the Mellin integral 239 4. The case of a function regular at infinity 244

3. Solving Problems for Linear Differential Equations by the Operational Method 246

1. Ordinary differential equations 247 2. Heat-conduction equation 251 3. The boundary-value problem for a partial differential equation 253

Appendix I. SADDLE-POINT METHOD 256

1. Introductory remarks 256 2. Laplace's method 259 3. The saddle-point method 266

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Appendix 11. THE WIENER-HOPF METHOD 275 1. Introductory remarks 275 2. Analytic properties о! the Fourier transiormation 279 3. Integral equations with a difference kernel 282 4. General scheme of the Wiener-Hopf method 287 5. Problems which reduce to integral equations with a difference

kernel 292 5.1. Derivation of Milne's equation 292 5.2. Investigating the solution of Milne's equation 296 5.3. Diffraction on a flat screen 300

6. Solving boundary-value problems ior partial differential equa­tions by the Wiener-Hopf method 301

Bibliography 306 Name Index 307 Subject Index ' 308