* 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.

**John P. D''Angelo**, University of Illinois, Urbana, IL

*Preface*

**Chapter 1. From the Real Numbers to the Complex Numbers**

1. Introduction

2. Number systems

3. Inequalities and ordered elds

4. The complex numbers

5. Alternative denitions of C

6. A glimpse at metric spaces

Chapter 2. Complex Numbers

1. Complex conjugation

2. Existence of square roots

3. Limits

4. Convergent innite series

5. Uniform convergence and consequences

6. The unit circle and trigonometry

7. The geometry of addition and multiplication

8. Logarithms

Chapter 3. Complex Numbers and Geometry

1. Lines, circles, and balls

2. Analytic geometry

3. Quadratic polynomials

4. Linear fractional transformations

5. The Riemann sphere

**Chapter 4. Power Series Expansions**

1. Geometric series

2. The radius of convergence

3. Generating functions

4. Fibonacci numbers

5. An application of power series

6. Rationality

**Chapter 5. Complex Dierentiation**

1. Denitions of complex analytic function

2. Complex dierentiation

3. The Cauchy-Riemann equations

4. Orthogonal trajectories and harmonic functions

5. A glimpse at harmonic functions

6. What is a dierential form?

**Chapter 6. Complex Integration**

1. Complex-valued functions

2. Line integrals

3. Goursat's proof

4. The Cauchy integral formula

5. A return to the denition of complex analytic function

**Chapter 7. Applications of Complex Integration**

1. Singularities and residues

2. Evaluating real integrals using complex variables methods

3. Fourier transforms

4. The Gamma function

**Chapter 8. Additional Topics**

1. The minimum-maximum theorem

2. The fundamental theorem of algebra

3. Winding numbers, zeroes, and poles

4. Pythagorean triples

5. Elementary mappings

6. Quaternions

7. Higher-dimensional complex analysis

Further reading

Bibliography

*Index *