About the book

The book covers basic aspects of complex numbers, complex variables and complex functions. It also deals with analytic functions, Laurent series etc.

Contents

Introduction 9
Chapter 1. THE COMPLEX VARIABLE AND FUNCTIONS OF A COMPLEX VARIABLE 11

1.1. Complex Numbers and Operations on Complex Numbers 11
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 19

1.3. The Concept of a Function of a Complex Variable. Continuity 20
a. Basic definitions 20
b. Continuity 23
c. Examples 26

1.4. Differentiating 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

1.7. 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
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 Analytic Function 76
a. Zeros of an analytic function 76
b. Uniqueness theorem 77

Chapter 3. ANALYTIC CONTINUATION. ELEMENTARY FUNCTIONS OF A COMPLEX VARIABLE 80

3.1. 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. Continuation across a boundary 100
d. Examples in constructing analytic continuations. Continuation 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
b. Expansion of an analytic function in a Laurent series 115

4.2. A Classification of the Isolated Singular Points of a Single-Valued Analytic Function 118
Chapter 5. RESIDUES AND THEIR APPLICATIONS 125

5.1. The Residue of an Analytic Function at an Isolated Singularity 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
a. Integrals of the form $\int^{2 \pi}_{0}R (\cos \theta \sin \theta ) d \theta$ 131
b. Integrals of the form $\int^{\infty}_{\infty} f(x)dx$ 132
c. Integrals of the form $\int^{\infty}_{\infty} \exp(iax)f(x)dx$. Jordan's lemma  135
d. The case of multiple-valued functions 141

5.3. Logarithmic Residue 147
a. The concept of a logarithmic residue 147
b. Counting the number of zeros of an analytic function 149

Chapter 6. CONFORMAL MAPPING 153

6.1. General Properties 153
a. Definition of a conformal mapping 153
b. Elementary examples 157
c. Basic principles 160
d. Riemann's theorem 166

6.2. Linear-Fractional Function 169

6.3. Zhukovsky's Function 179

6.4. Schwartz-Christoffel Integral. Transformation of Polygons 181

Chapter 7. ANALYTIC FUNCTIONS IN THE SOLUTION OF BOUNDARY-VALUE PROBLEMS 191

7. 1. Generalities 191
a. The relationship of analytic and harmonic functions 191
b. Preservation of the Laplace operator in a conformal mapping 192
c. Dirichlet's problem 194
d. Constructing a source function 197

7.2. Applications to Problems in Mechanics and Physics 199
a. Two-dimensional steady-state flow of a fluid 199
b. A two-dimensional electrostatic field 211

Chapter 8. FUNDAMENTALS OF OPERATIONAL CALCULUS 221

8.1. Basic Properties of the Laplace Transformation 221
a. Definition 221
b. Transforms of elementary functions 225
c. Properties of a transform 227
d. Table of properties of transforms 236
e. Table of transforms 236
8.2. Determining the Original Function from the Transform 238
a. Mellin's formula 238
h. Existence conditions of the original function 241
c. Computing the Mellin integral 245
d. The case of a function regular at infinity 249
8.3. Solving Problems for Linear Differential Equations by the Operational Method 252
a. Ordinary differential equations 252
b. Heat-conduction equation 257
c. The boundary-value problem for a partial differential equation 259
Appendix I. SADDLE-POINT METHOD 261
I.1. Introductory Remarks 261

I.2. Laplace's Method 264
I.3. The Saddle-Point Method 271
Appendix II. THE WIENER-HOPF METHOD 280

II.1. Introductory Remarks 280
11.2. Analytic Properties of the Fourier Transformation 284
11.3. Integral Equations with a Difference Kernel 287
II.4. General Scheme of the Wiener-Hopf Method 292
II.5. Problems Which Reduce to Integral Equations with a Difference
Kernel 297
a. Derivation of Milne's equation 297
b. Investigating the solution of Milne's equation 301
c. Diffraction on a flat screen 305
II.6. Solving Boundary-Value Problems for Partial Differential Equations by the Wiener-Hopf Method 306

Appendix III. FUNCTIONS OF MANY COMPLEX VARIABLES 310

III.1. Basic Definitions 310
III.2. The Concept of an Analytic Function of Many Complex Variables 311
III.3. Cauchy's Formula 312
III.4. Power Series 314
III.5. Taylor's Series 316
III.6. Analytic Continuation 317

Appendix IV. WATSON'S METHOD 320
References 328
Name Index 329
Subject Index 330

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