## Lectures on the Theory of Functions of a Complex Variable – Sidorov, Fedoryuk, Shabunin

In this post we will see the book Lectures on the Theory of Functions of a Complex Variable by Yu. V. Sidorov, M. V. Fedoryuk, M. I. Shabunin.

## About the book

This book is based on more than ten years experience in teaching the theory of functions of a complex variable at the Moscow Physics and Technology Institute. It is a textbook for students of universities and institutes of technology with an advanced mathematical program.
We believe that it can also be used for independent study.
We have stressed the methods of the theory that are often used in applied sciences. These methods include series expansions, conformal mapping, application of the theory of residues to evaluating definite integrals, and asymptotic methods. The material is structured in a way that will give the reader the maximum assistance in mastering the basics of the theory. To this end we have provided a wide range of worked-out examples. We hope that these will help the reader
acquire a deeper understanding of the theory and experience in
problem solving.

The book was translated from Russian by Eugene Yankovsky and was first published by Mir in 1985.

PDF | OCR | 600 dpi | 25 MB | Bookmarked

You can get the book here.

## Contents

Preface 5
Chapter I Introduction 9

1 Complex Numbers 9
2 Sequences and Series of Complex Numbers 20
3 Curves and Domains in the Complex Plane 25
4 Continuous Functions of a Complex Variable 36
5 Integrating Functions of a Complex Variable 45
6 The Function arg z 51
Chapter II Regular Functions 59

7 Differentiable Functions. The Cauchy-Riemann Equations 59
8 The Geometric Interpretation of the Derivative 66
9 Cauchy’s Integral Theorem 76 .–
10 Cauchy’s Integral Formula 84
11 Power Series 87
12 Properties of Regular Functions 90
13 The Inverse Function 102
14 The Uniqueness Theorem 108
15 Analytic Continuation 110
16 Integrals Depending on a Parameter 112
Chapter III The Laurent Series. Isolated Singular Points of a Single-Valued Functions 123
17 The Laurent Series 123
18 Isolated Singular Points of Single- Valued Functions 128
19 Liouville’s Theorem 138
Chapter IV Multiple-Valued Analytic Functions 141

20 The Concept of an Analytic Function 141
21 The Function In z 147
22 The Power Function. Branch Points of Analytic Functions 155
23 The Primitive of an Analytic Function. Inverse Trigonometric
Functions 166
24 Regular Branches of Analytic Functions 170
25 Singular Boundary Points 189
26 Singular Points of Analytic Functions. The Concept of a Riemann
Surface 194
27 Analytic Theory of Linear Second-Order Ordinary Differential
Equations 204

Chapter V Residues and Their Applications 220
28 Residue Theorems 220
29 Use of Residues for Evaluating Definite Integrals 230
30 The Argument Principle and Rouche’s Theorem 255
31 The Partial-Fraction Expansion of Meromorphic Functions 260
Chapter VI Conformal Mapping 270

32 Local Properties of Mappings Performed by Regular Functions 270
33 General Properties of Conformal Mappings 276
34 The Linear-Fractional Function 282
35 Conformal Mapping Performed by Elementary Functions 291
36 The Riemann-Schwarz Symmetry Principle 315
37 The Schwarz-Cristoffel Transformation Formula 326
38 The Dirichlet Problem 339
39 Vector Fields in a Plane 354
40 Some Physical Problems from Vector Field Theory 363
Chapter VII Simple Asymptotic Methods 371

41 Some Asymptotic Estimates 371
42 Asymptotic Expansions 389
43 Laplace’s Method 396
44 The Method of Stationary Phase 409
45 The Saddle-Point Method ‘•18
46 Laplace’s Method of Contour Integration 434

Chapter VIII Operational Calculus 446

47 Basic Properties of the Laplace Transformation 446
48 Reconstructing Object Function from Result Function 454
49 Solving Linear Differential Equations via the Laplace Transformation
468
50 String Vibrations from Instantaneous Shock 476
Selected Bibliography 486
Name Index 488
Subject Index 489