Intro
Preface to Second Edition
Preface to the First Edition
Contents
Part I Complex Analysis
1 Complex Numbers
1.1 Our Number System
1.1.1 Addition and Multiplication of Integers
1.1.2 Inverse Operations
1.1.3 Negative Numbers
1.1.4 Fractional Numbers
1.1.5 Irrational Numbers
1.1.6 Imaginary Numbers
1.2 Logarithm
1.2.1 Napier's Idea of Logarithm
1.2.2 Briggs' Common Logarithm
1.3 A Peculiar Number Called e
1.3.1 The Unique Property of e
1.3.2 The Natural Logarithm
1.3.3 Approximate Value of e

1.4 The Exponential Function as an Infinite Series
1.4.1 Compound Interest
1.4.2 The Limiting Process Representing e
1.4.3 The Exponential Function ex
1.5 Unification of Algebra and Geometry
1.5.1 The Remarkable Euler Formula
1.5.2 The Complex Plane
1.6 Polar Form of Complex Numbers
1.6.1 Powers and Roots of Complex Numbers
1.6.2 Trigonometry and Complex Numbers
1.6.3 Geometry and Complex Numbers
1.7 Elementary Functions of Complex Variable
1.7.1 Exponential and Trigonometric Functions of z
1.7.2 Hyperbolic Functions of z

1.7.3 Logarithm and General Power of z
1.7.4 Inverse Trigonomeric and Hyperbolic Functions
2 Complex Functions
2.1 Analytic Functions
2.1.1 Complex Function as Mapping Operation
2.1.2 Differentiation of a Complex Function
2.1.3 Cauchy-Riemann Conditions
2.1.4 Cauchy-Riemann Equations in Polar Coordinates
2.1.5 Analytic Function as a Function of z Alone
2.1.6 Analytic Function and Laplace's Equation
2.2 Complex Integration
2.2.1 Line Integral of a Complex Function
2.2.2 Parametric Form of Complex Line Integral
2.3 Cauchy's Integral Theorem

2.3.1 Green's Lemma
2.3.2 Cauchy-Goursat Theorem
2.3.3 Fundamental Theorem of Calculus
2.4 Consequences of Cauchy's Theorem
2.4.1 Principle of Deformation of Contours
2.4.2 The Cauchy Integral Formula
2.4.3 Derivatives of Analytic Function
3 Complex Series and Theory of Residues
3.1 A Basic Geometric Series
3.2 Taylor Series
3.2.1 The Complex Taylor Series
3.2.2 Convergence of Taylor Series
3.2.3 Analytic Continuation
3.2.4 Uniqueness of Taylor Series
3.3 Laurent Series
3.3.1 Uniqueness of Laurent Series
3.4 Theory of Residues

3.4.1 Zeros and Poles
3.4.2 Definition of the Residue
3.4.3 Methods of Finding Residues
3.4.4 Cauchy's Residue Theorem
3.4.5 Second Residue Theorem
3.5 Evaluation of Real Integrals with Residues
3.5.1 Integrals of Trigonometric Functions
3.5.2 Improper Integrals I: Closing the Contour with a Semicircle at Infinity
3.5.3 Fourier Integral and Jordan's Lemma
3.5.4 Improper Integrals II: Closing the Contour with Rectangular and Pie-Shaped Contour
3.5.5 Integration Along a Branch Cut
3.5.6 Principal Value and Indented Path Integrals
4 Conformal Mapping