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