Complex Analysis

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

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