The text covers a broad spectrum between basic and advanced **complex variables** on the one hand and between theoretical and applied or computational material on the other hand. With careful selection of the emphasis put on the various sections, examples, and exercises, the book can be used in a one- or two-semester course for undergraduate mathematics majors, a one-semester course for engineering or physics majors, or a one-semester course for first-year mathematics graduate students. It has been tested in all three settings at the University of Utah.

The exposition is clear, concise, and lively. There is a clean and modern approach to Cauchy's theorems and Taylor series expansions, with rigorous proofs but no long and tedious arguments. This is followed by the rich harvest of easy consequences of the existence of power series expansions.

Through the central portion of the text, there is a careful and extensive treatment of residue theory and its application to computation of integrals, conformal mapping and its applications to applied problems, analytic continuation, and the proofs of the Picard theorems.

Chapter 8 covers material on infinite products and zeroes of entire functions. This leads to the final chapter which is devoted to the Riemann zeta function, the Riemann Hypothesis, and a proof of the Prime Number Theorem.

Joseph L. Taylor, *University** of Utah, Salt Lake City, UT*

*Preface*

**Chapter 1. The Complex Numbers**

1.1. Definition and Simple Properties

1.2. Convergence in C

1.3. The Exponential Function

1.4. Polar Form for Complex Numbers

**Chapter 2. Analytic Functions**

2.1. Continuous Functions

2.2. The Complex Derivative

2.3. Contour Integrals

2.4. Properties of Contour Integrals

2.5. Cauchy's Integral Theorem for a Triangle

2.6. Cauchy's Theorem for a Convex Set

2.7. Properties of the Index Function

**Chapter 3. Power Series Expansions**

3.1. Uniform Convergence

3.2. Power Series Expansions

3.3. Liouville's Theorem

3.4. Zeroes and Singularities

3.5. The Maximum Modulus Principle

**Chapter 4. The General Cauchy Theorems**

4.1. Chains and Cycles

4.2. Cauchy's Theorems

4.3. Laurent Series

4.4. The Residue Theorem

4.5. RouchÂ´e's Theorem and Inverse Functions

4.6. Homotopy

**Chapter 5. Residue Theory**

5.1. Computing Residues

5.2. Evaluating Integrals Using Residues

5.3. Fourier Transforms

5.4. The Laplace and Mellin Transforms

5.5. Summing Infinite Series

**Chapter 6. Conformal Mappings**

6.1. Definition and Examples

6.2. The Riemann Sphere

6.3. Linear Fractional Transformations

6.4. The Riemann Mapping Theorem

6.5. The Poisson Integral

6.6. The Dirichlet Problem

Chapter 7. Analytic Continuation and the Picard Theorems

Chapter 7. Analytic Continuation and the Picard Theorems

7.1. The Schwarz Reflection Principle

7.2. Continuation Along a Curve

7.3. Analytic Covering Maps

7.4. The Picard Theorems

**Chapter 8. Infinite Products**

8.1. Convergence of Infinite Products

8.2. Weierstrass Products

8.3. Entire Functions of Finite Order

8.4. Hadamard's Factorization Theorem

**Chapter 9. The Gamma and Zeta Functions**

9.1. Euler's Gamma Function

9.2. The Riemann Zeta Function

9.3. Properties of ?

9.4. The Riemann Hypothesis and Prime Numbers

9.5. A Proof of the Prime Number Theorem

Bibliography

*Index*