## Description

This textbook is meant for students of first course on Complex Analysis. The book provides balanced coverage of both theory and practice of concepts. A wide variety of problems and lucid language will help students understand and apply the concepts easily. The step-by-step solutions to solved problems enables clear understanding. A variety of solved and unsolved problems have been incorporated to help students ace university and competitive examinations.

## Table of Content

**Chapter 1 Complex Numbers**

1.1 Introduction

1.2 Complex Numbers

1.3 Modulus Amplitude Form

1.4 Procedure for Finding the Principal Value of θ

1.5 Geometrical Representation of Complex Numbers

1.5.1 Representation of (z1 + z2)

1.5.2 Representation of (z1 − z2)

1.5.3 Representation of z1z2

1.5.4 Representation of (z1/z2)

1.6 De Moivre’s Theorem

**Chapter 2 Analytic Functions**

2.1 The Complex Variable

2.2 Function of a Complex Variable

2.2.1 Limit of a Function of a Complex Variable

2.2.2 Continuity of f(z)

2.2.3 Derivative of f(z)

2.3 Analytic Function

2.3.1 Cauchy−Riemann Equations

2.3.2 C.R. Equations in Polar Co-ordinates

2.4 Properties of Analytic Functions

2.4.1 Property 1

2.4.2 Property 2

2.4.3 Property 3

2.4.4 Property 4

2.5 Construction of an Analytic Function, When its Real or

Imaginary Part is Known

2.5.1 Method 1

2.5.2 Method 2 (Milne−Thomson Method)

2.5.3 Applications

**Chapter 3 Conformal Mapping**

3.1 Mapping

3.2 Conformal Mapping

3.2.1 Definition

3.2.2 Theorem

3.3 Some Simple Transformations

3.3.1 Translation

3.3.2 Magnification

3.3.3 Magnification and Rotation

3.3.4 Magnification, Rotation and Translation

3.3.5 Inversion and Reflection

3.4 Some Standard Transformations

3.4.1 The Transformation w = z2

3.4.2 The Transformation w = ez

3.4.3 The Transformation w = sinz

3.4.4 The Transformation w = coshz

3.4.5 The Transformation w = z + k^2/z, where k is Real and Positive

3.5 Bilinear and Schwarz−Christoffel Transformations

3.5.1 Bilinear Transformation

3.5.2 Schwarz−Christoffel Transformation

**Chapter 4 Complex Integration**

4.1 Introduction

4.2 Simply and Multiply Connected Regions

4.3 Cauchy’s Integral Theorem or Cauchy’s Fundamental Theorem

4.3.1 Extension of Cauchy’s Integral Theorem

4.4 Cauchy’s Integral Formula

4.4.1 Extension of Cauchy’s Integral Formula to a Doubly

Connected Region

4.4.2 Cauchy’s Integral Formulas for the Derivatives of an

Analytic Function

**Chapter 5 Expansions, Singularities and Residues**

5.1 Series Expansions of Functions of Complex

Variable-Power Series

5.1.1 Power Series

5.1.2 Taylor’s Series (Taylor’s Theorem)

5.2 Laurent’s Series (Laurent’s Theorem)

5.3 Classification of Singularities

5.3.1 Isolated Singularity

5.3.2 Pole

5.3.3 Essential Singularity

5.3.4 Removable Singularity

5.4 Residues and Evaluation of Residues

5.4.1 Formulas for the Evaluation of Residues

5.5 Cauchy’s Residue Theorem

**Chapter 6 Contour Integration**

6.1 Contour Integration – Evaluation of Real Integrals

6.2 Type 1

6.2.1 Cauchy’s Lemma

6.3 Type 2

6.3.1 Jordan’s Lemma

6.4 Type 3

## About The Author

**T. Veerarajan** is Dean (Retd.), Department of Mathematics, Velammal College of Engineering and Technology, Madurai, Tamil Nadu. A Gold Medalist from Madras University, he has had a brilliant academic career all through. He has 50 years of teaching experience at undergraduate and postgraduate levels in various established engineering colleges in Tamil Nadu including Anna University, Chennai.

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