Applied Complex Variables

Cover; Title Page; Copyright Page; Preface; Contents; Part I. Analytic Function Theory; Chapter 1. The Complex Number Plane; 1.1 Introduction; 1.2 Complex Numbers; 1.3 The Complex Plane; 1.4 Point Sets in the Plane; 1.5 Stereographic Projection. The Extended Complex Plane; 1.6 Curves and Regions; Chapter 2. Functions of a Complex Variable; 2.1 Functions and Limits; 2.2 Differentiability and Analyticity; 2.3 The Cauchy-Riemann Conditions; 2.4 Linear Fractional Transformations; 2.5 Transcendental Functions; 2.6 Riemann Surfaces; Chapter 3. Integration in the Complex Plane; 3.1 Line Integrals.

3.2 The Definite Integral3.3 Cauchy's Theorem; 3.4 Implications of Cauchy's Theorem; 3.5 Functions Defined by Integration; 3.6 Cauchy Formulas; 3.7 Maximum Modulus Principle; Chapter 4. Sequences and Series; 4.1 Sequences of Complex Numbers; 4.2 Sequences of Complex Functions; 4.3 Infinite Series; 4.4 Power Series; 4.5 Analytic Continuation; 4.6 Laurent Series; 4.7 Double Series; 4.8 Infinite Products; 4.9 Improper Integrals; 4.10 The Gamma Function; Chapter 5. Residue Calculus; 5.1 The Residue Theorem; 5.2 Evaluation of Real Integrals; 5.3 The Principle of the Argument.

5.4 Meromorphic Functions5.5 Entire Functions; Part II. Applications of Analytic Function Theory; Chapter 6. Potential Theory; 6.1 Laplace's Equation in Physics; 6.2 The Dirichlet Problem; 6.3 Green's Functions; 6.4 Conformal Mapping; 6.5 The Schwarz-Christoffel Transformation; 6.6 Flows with Sources and Sinks; 6.7 Volume and Surface Distributions; 6.8 Singular Integral Equations; Chapter 7. Ordinary Differential Equations; 7.1 Separation of Variables; 7.2 Existence and Uniqueness Theorems; 7.3 Solution of a Linear Second-Order Differential Equation Near an Ordinary Point.

7.4 Solution of a Linear Second-Order Differential Equation Near a Regular Singular Point7.5 Bessel Functions; 7.6 Legendre Functions; 7.7 Sturm-Liouville Problems; 7.8 Fredholm Integral Equations; Chapter 8. Fourier Transforms; 8.1 Fourier Series; 8.2 The Fourier Integral Theorem; 8.3 The Complex Fourier Transform; 8.4 Properties of the Fourier Transform; 8.5 The Solution of Ordinary Differential Equations; 8.6 The Solution of Partial Differential Equations; 8.7 The Solution of Integral Equations; Chapter 9. Laplace Transforms; 9.1 From Fourier to Laplace Transform.

9.2 Properties of the Laplace Transform9.3 Inversion of Laplace Transforms; 9.4 The Solution of Ordinary Differential Equations; 9.5 Stability; 9.6 The Solution of Partial Differential Equations; 9.7 The Solution of Integral Equations; Chapter 10. Asymptotic Expansions; 10.1 Introduction and Definitions; 10.2 Operations on Asymptotic Expansions; 10.3 Asymptotic Expansion of Integrals; 10.4 Asymptotic Solutions of Ordinary Differential Equations; References; Index.

Analytic function theory is a traditional subject going back to Cauchy and Riemann in the 19th century. Once the exclusive province of advanced mathematics students, its applications have proven vital to today's physicists and engineers. In this highly regarded work, Professor John W. Dettman offers a clear, well-organized overview of the subject and various applications - making the often-perplexing study of analytic functions of complex variables more accessible to a wider audience. The first half of Applied Complex Variables, designed for sequential study, is a step-by-step treatment of fun.