A First Course in Complex Analysis (Synthesis Lectures on Mathematics and Statistics)

Preface
Acknowledgments
Basics of Complex Numbers
	Introduction
		Cartesian and Polar Forms
		Addition and Multiplication of Complex Numbers
		Exercises
	The Exponential Function
		Euler's Formula
		The Exponential as Polar Form
		Exercises
	Conversion between Cartesian and Polar Forms
		Exercises
	Conjugation
		Exercises
	Integer and Rational Powers
		Exercises
	Stereographic Projection
		Exercises
Functions of a Complex Variable
	Set Terminology
	Single-Valued and Multi-Valued Functions
		Exercises
	Lines and Circles
		Elementary Mappings of Lines and Circles
		Exercises
	Visualizing Complex Functions
	Some Elementary Functions
		Polynomials
		Rational Functions
		Rational Powers
		The Exponential
		Trigonometric Functions
		Hyperbolic Functions
		The Logarithmic Function
		Complex Powers
		Inverse Trigonometric Functions
		Inverse Hyperbolic Functions
		Exercises
Differentiation
	The Derivative
		Exercises
	Geometric Interpretation of the Derivative
		Exercises
	The Cauchy–Riemann Equations
		Sufficient Conditions for Differentiability
		Other Forms of the Cauchy–Riemann Equations
		Exercises
	Analytic Functions
		Invertibility
		Harmonic Functions
		Exercises
	Singular Points
		Isolated Singularities
		Branch Points
		Other Singularities
		Exercises
	Riemann Surfaces
Contour Integration
	Arcs, Contours, and Parameterizations
		Definite Integrals and Derivatives of Parameterizations
		An Application: Fourier Series
		Contours
		Exercises
	Contour Integrals
		Exercises
Cauchy Theory
	The Cauchy–Goursat Theorem and its Consequences
		Path Independence
		Complex Extension of the Fundamental Theorem of Calculus
		Path Deformation
		Exercises
	The Cauchy Integral Formulas and their Consequences
		Morera's Theorem
		Cauchy's Inequality
		Liouville's Theorem
		Fundamental Theorem of Algebra
		Gauss' Mean Value Theorem
		Maximum Modulus Theorem
		Minimum Modulus Theorem
		Poisson's Integral Formulas for the Circle and Half-Plane
		Exercises
	Counting Zeros and Poles
		Argument Theorem
		Rouché's Theorem
		Argument Principle
		Exercises
Series
	Convergence
		Sequences
		Series
		Series Convergence Tests
		Uniform Convergence Results
		Exercises
	Power Series
	Taylor Series
		Zeros of Analytic Functions
		Analytic Continuation
		Exercises
	Laurent Series
		Exercises
	Isolated Singularities Again
		Exercises
Residues
	Calculation of Residues
		Exercises
	The Residue Theorem
		Exercises
	Calculation of Certain Real Integrals
		Integrals of the Form _02 F(cos,sin) d
		Improper Real Integrals
		Exercises
Conformal Mapping
	Conformal Maps
		Exercises
	Application to Laplace's Equation
		Exercises
Greek Alphabet
Answers to Selected Exercises
Author's Biography
Index