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Table of Contents
1 Complex Numbers
Sums and Products
Basic Algebraic Properties
Further Properties
Moduli
Complex Conjugates
Exponential Form
Products and Quotients in Exponential Form
Roots of Complex Numbers
Examples
Regions in the Complex Plane
2 Analytic Functions
Functions of a Complex Variable
Mappings
Mappings by the Exponential Function
Limits
Theorems on Limits
Limits Involving the Point at Infinity
Continuity
Derivatives
Differentiation Formulas
Cauchy–Riemann Equations
Sufficient Conditions for Differentiability
Polar Coordinates
Analytic Functions
Examples
Harmonic Functions
Uniquely Determined Analytic Functions
Reflection Principle
3 Elementary Functions
The Exponential Function
The Logarithmic Function
Branches and Derivatives of Logarithms
Some Identities Involving Logarithms
Complex Exponents
Trigonometric Functions
Hyperbolic Functions
Inverse Trigonometric and Hyperbolic Functions
4 Integrals
Derivatives of Functions w(t)
Definite Integrals of Functions w(t)
Contours
Contour Integrals
Examples
Upper Bounds for Moduli of Contour Integrals
Antiderivatives
Examples
Cauchy–Goursat Theorem
Proof of the Theorem
Simply and Multiply Connected Domains
Cauchy Integral Formula
Derivatives of Analytic Functions
Liouville’s Theorem and the Fundamental Theorem of Algebra
Maximum Modulus Principle
5 Series
Convergence of Sequences
Convergence of Series
Taylor Series
Examples
Laurent Series
Examples
Absolute and Uniform Convergence of Power Series
Continuity of Sums of Power Series
Integration and Differentiation of Power Series
Uniqueness of Series Representations
Multiplication and Division of Power Series
6 Residues and Poles
Residues
Cauchy’s Residue Theorem
Using a Single Residue
The Three Types of Isolated Singular Points
Residues at Poles
Examples
Zeros of Analytic Functions
Zeros and Poles
Behavior of f Near Isolated Singular Points
7 Applications of Residues
Evaluation of Improper Integrals
Example
Improper Integrals from Fourier Analysis
Jordan’s Lemma
Indented Paths
An Indentation Around a Branch Point
Integration Along a Branch Cut
Definite Integrals Involving Sines and Cosines
Argument Principle
Rouché’s Theorem
Inverse Laplace Transforms
Examples
8 Mapping by Elementary Functions
Linear Transformations
The Transformation w = 1/z
Mappings by 1/z
Linear Fractional Transformations
An Implicit Form
Mappings of the Upper Half Plane
The Transformation w = sin z
Mappings by z2 and Branches of z1/2
Square Roots of Polynomials
Riemann Surfaces
Surfaces for Related Functions
9 Conformal Mapping
Preservation of Angles
Scale Factors
Local Inverses
Harmonic Conjugates
Transformations of Harmonic Functions
Transformations of Boundary Conditions
10 Applications of Conformal Mapping
Steady Temperatures
Steady Temperatures in a Half Plane
A Related Problem
Temperatures in a Quadrant
Electrostatic Potential
Potential in a Cylindrical Space
Two-Dimensional Fluid Flow
The Stream Function
Flows Around a Corner and Around a Cylinder
11 The Schwarz–Christoffel Transformation
Mapping the Real Axis onto a Polygon
Schwarz–Christoffel Transformation
Triangles and Rectangles
Degenerate Polygons
Fluid Flow in a Channel Through a Slit
Flow in a Channel with an Offset
Electrostatic Potential about an Edge of a Conducting Plate