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 fundamentals, presenting superior coverage of concepts of complex analysis, including the complex number plane; functions and limits; the Cauchy-Riemann conditions for differentiability; Riemann surfaces; the definite integral; power series; meromorphic functions; and much more. The second half provides lucid exposition of five important applications of analytic function theory, each approachable independently of the others: potential theory; ordinary differential equations; Fourier transforms; Laplace transforms; and asymptotic expansions. Helpful exercises are included at the end of each topic in every chapter.
The two-part structure of Applied Complex Variables affords the college instructor maximum classroom flexibility. Once fundamentals are mastered, applications can be studied in any sequence desired. Depending on how many are selected for study, Professor Dettman's impressive text is ideal for either a one- or two-semester course. And, of course, the ambitious student possessing a knowledge of basic calculus will find its straightforward approach rewarding to his independent study efforts.
Applied Complex Variables is a cogent, well-written introduction to an important and exciting branch of advanced mathematics - serving both the theoretical needs of the mathematics specialist and the applied math needs of the physicist and engineer. Students and teachers alike will welcome this timely, moderately priced reissue of a widely respected work.
Reihe
Sprache
Verlagsgruppe
Dateigröße
ISBN-13
978-0-486-15828-0 (9780486158280)
Schweitzer Klassifikation
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 Integral
3.3 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 Functions
5.5 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 Point
7.5 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 Transform
9.3 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
