Complex Analysis: Theory and Applications (De Gruyter Textbook)

✍ Scribed by Teodor Bulboacǎ, Santosh B. Joshi, Pranay Goswami

• Publisher: De Gruyter
• Year: 2019
• Tongue: English
• Leaves: 423
• Edition: 1
• Category: Library

✦ Synopsis

This book is an in-depth and modern presentation of important classical results in complex analysis and is suitable for a first course on the topic, as taught by the authors at several universities. The level of difficulty of the material increases gradually from chapter to chapter, and each chapter contains many exercises with solutions and applications of the results, with the particular goal of showcasing a variety of solution techniques.

✦ Table of Contents

Contents
Dedication
Preface
1 Complex numbers
1.1 The field of the complex numbers
1.2 The complex plane
1.3 The topological and metric structure of the complex plane
1.3.1 Basic definitions and notation
1.4 Complex function, limits, continuity
1.5 The compactified complex plane
1.5.1 The geometric interpretation of the φ function
1.5.2 The topological structure of ℂ
1.6 Exercises
2 Holomorphic functions
2.1 The derivative of the real valued complex functions
2.2 The differentiability of a complex function
2.3 The derivative of a complex function
2.3.1 The properties of the derivative
2.4 The geometric interpretation of the derivative
2.5 Entire functions
2.5.1 The polynomial function
2.5.2 The exponential function
2.5.3 Complex trigonometric functions
2.5.4 Complex hyperbolic functions
2.6 Bilinear transforms
2.6.1 Decompositions in elementary functions
2.7 The Möbius-type groups
2.8 Multivalued functions
2.8.1 The logarithmic function
2.8.2 Inverse trigonometric functions
2.8.3 The power function
2.9 Exercises
2.9.1 Real variable complex functions
2.9.2 The derivative of a complex function
2.9.3 Entire functions
2.9.4 Bilinear transforms
3 The complex integration
3.1 The homotopic theory of the paths
3.1.1 Simply connected domains
3.1.2 Functions of bounded variation and paths
3.2 The complex integral
3.2.1 The Riemann–Stieltjes integral for complex valued functions
3.3 The Cauchy theorem
3.3.1 The connection between the integral and the primitive function
3.3.2 The Cauchy theorem
3.4 The Cauchy formula for the disc
3.5 The analytical branches of multivalued functions
3.6 The index of a path (curve) with respect to a point
3.7 Cauchy formula for closed curves
3.8 Some consequences of Cauchy formula
3.9 Schwarz and Poisson formulas
3.10 Exercises
3.10.1 The complex integral
3.10.2 The Cauchy theorem
3.10.3 The Cauchy formula for the disc
3.10.4 Some consequences of Cauchy formula
3.10.5 Multivalued functions analytical branches
4 Sequences and series of holomorphic functions
4.1 Sequences of holomorphic functions
4.2 Series of functions
4.3 Power series
4.4 The analyticity of holomorphic functions
4.5 The zeros of holomorphic functions
4.6 The maximum principle of the holomorphic functions
4.7 Laurent series
4.8 Isolated singular points
4.9 Meromorphic functions
4.10 Exercises
4.10.1 Power series
4.10.2 Taylor and Laurent series
4.10.3 Isolated singular points
4.10.4 The module maximum of the holomorphic functions
5 Residue theory
5.1 Residue theorem
5.2 Applications of the residue theorem to the calculation of the integrals
5.3 The study of meromorphic functions with the residue theorem
5.4 Exercises
5.4.1 Residue theorem
5.4.2 Applications of the residue theorem to the calculation of the trigonometric integrals
5.4.3 Applications of the residue theorem to the calculation of the improper integrals
5.4.4 The study of meromorphic functions using the residue theorem
6 Conformal representations
6.1 Special classes of holomorphic functions
6.2 Univalent functions
6.3 The problem of conformal representation
6.4 The Riemann mapping theorem
6.5 Exercises
7 Solutions to the chapterwise exercises
Solutions to the exercises of Chapter 1
Solutions to the exercises of Chapter 2
Solutions to the exercises of Chapter 3
Solutions to the exercises of Chapter 4
Solutions to the exercises of Chapter 5
Solutions to the exercises of Chapter 6
Bibliography
Index