A First Course in Complex Analysis

✍ Scribed by Allan R. Willms

• Publisher: Morgan & Claypool
• Year: 2022
• Tongue: English
• Leaves: 237
• Series: Synthesis Lectures on Mathematics and Statistics
• Category: Library

✦ Synopsis

This book introduces complex analysis and is appropriate for a first course in the subject at typically the third-year University level . It introduces the exponential function very early but does so rigorously. It covers the usual topics of functions, differentiation, analyticity, contour integration, the theorems of Cauchy and their many consequences, Taylor and Laurent series, residue theory, the computation of certain improper real integrals, and a brief introduction to conformal mapping. Throughout the text an emphasis is placed on geometric properties of complex numbers and visualization of complex mappings.

✦ Table of Contents

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