An Introduction to Complex Analysis and Geometry provides the reader with a deep appreciation of complex analysis and how this subject fits into mathematics. The book developed from courses given in the Campus Honors Program at the University of Illinois Urbana-Champaign. These courses aimed to share with students the way many mathematics and physics problems magically simplify when viewed from the perspective of complex analysis. The book begins at an elementary level but also contains advanced material.

The first four chapters provide an introduction to complex analysis with many elementary and unusual applications. Chapters 5 through 7 develop the Cauchy theory and include some striking applications to calculus. Chapter 8 glimpses several appealing topics, simultaneously unifying the book and opening the door to further study.

The 280 exercises range from simple computations to difficult problems. Their variety makes the book especially attractive.

A reader of the first four chapters will be able to apply complex numbers in many elementary contexts. A reader of the full book will know basic one complex variable theory and will have seen it integrated into mathematics as a whole. Research mathematicians will discover several novel perspectives.

Table of Contents

Preface

From the Real Numbers to the Complex Numbers
Introduction
Number systems
Inequalities and ordered fields
The complex numbers
Alternative definitions of C
A glimpse at metric spaces
Complex Numbers
Complex conjugation
Existence of square roots
Limits
Convergent infinite series
Uniform convergence and consequences
The unit circle and trigonometry
The geometry of addition and multiplication
Logarithms
Complex Numbers and Geometry
Lines, circles and balls
Analytic geometry
Quadratic polynomials
Linear fractional transformations
The Riemann sphere

Power Series Expansions
Geometric series
The radius of convergence
Generating functions
Fibonacci numbers
An application of power series
Rationality

Complex Differentiation
Definitions of complex analytic function
Complex differentiation
The Cauchy-Riemann equations
Orthogonal trajectories and harmonic functions
A glimpse at harmonic functions
What is a differential form?

Complex Integration
Complex-valued functions
Line integrals
Goursat’s proof
The Cauchy integral formula
A return to the definition of complex analytic function

Applications of Complex Integration
Singularities and residues
Evaluating real integrals using complex variables methods
Fourier transforms
The Gamma function

Additional Topics
The minimum-maximum theorem
The fundamental theorem of algebra
Winding numbers, zeroes, and poles
Pythagorean triples
Elementary mappings
Quaternions
Higher-dimensional complex analysis

Further reading
Bibliography
Index

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