Complex variables and applications

Title
Contents
1 Complex Numbers
	Sums and Products
	Basic Algebraic Properties
	Further Properties
	Vectors and Moduli
	Complex Conjugates
	Exponential Form
	Products and Powers in Exponential Form
	Arguments of Products and Quotients
	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
	Some Examples
	Examples with Branch Cuts
	Upper Bounds for Moduli of Contour Integrals
	Antiderivatives
	Proof of the Theorem
	Cauchy–Goursat Theorem
	Proof of the Theorem
	Simply Connected Domains
	Multiply Connected Domains
	Cauchy Integral Formula
	An Extension of the Cauchy Integral Formula
	Some Consequences of the Extension
	Liouville’s Theorem and the Fundamental Theorem of Algebra
	Maximum Modulus Principle
5 Series
	Convergence of Sequences
	Convergence of Series
	Taylor Series
	Proof of Taylor’s Theorem
	Examples
	Laurent Series
	Proof of Laurent’s Theorem
	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
	Isolated Singular Points
	Residues
	Cauchy’s Residue Theorem
	Residue at Infinity
	The Three Types of Isolated Singular Points
	Residues at Poles
	Examples
	Zeros of Analytic Functions
	Zeros and Poles
	Behavior of Functions 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
	Rouch´e’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
12 Integral Formulas of the Poisson Type
	Poisson Integral Formula
	Dirichlet Problem for a Disk
	Related Boundary Value Problems
	Schwarz Integral Formula
	Dirichlet Problem for a Half Plane
	Neumann Problems
Appendixes
	Bibliography
	Table of Transformations of Regions
Index