Preface
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
ProofofTaylor's Theorem
Examples
Laurent Series
ProofofLaurent's 111eorem
Examples
Absolute and Uniform Convergence of Power Series
Continuity of Sums of Power Series
Integration and Differentiation ofPower Series
Uniqueness of Series Representations
Multiplication and Division of Power Series
6 Residues and Poles
Isolated Singular Poims
Residues
Cauchy's Residue Theorem
Residue at Infinity
The Three Types of Isolated Singular Points
ResiduCS 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 P0int
Integration Along a Branch Cut
Definite Integrals Involving Sines and Cosines
Argument Principle
Rouch6's Theorem
Inverse Laplace Transforms
Examples
8 Mapping by Elementary Functions
Linear Transformations
The TransfoITnation w=1/Z
Mappings by 1/Z
Linear Fractional Transformations
An Implicit Form
Mappings ofthe Upper HalfPlane
The Transformation w=sinZ
Mappings by z2 and Branches of z1/2
Square Roots of Polynomials
Riemann Surfaces
Surfaces forRelatedFuncfions
9 Conformal Mapping
10 Applications of Conformal Mapping
11 The Schwarz-Chrstoffer Transformation
12 Integral Formulas of the Poisson Type
Appendixes
Index