Applied complex variable Dover publications inc (Dover Publications Inc.)

Part I. Analytic Function TheoryChapter 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 RegionsChapter 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 SurfacesChapter 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 żeby Integration 3.6 Cauchy Formulas 3.7 Maximum Modulus PrincipleChapter 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 FunctionChapter 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 FunctionsPart II. Applications of Analytic Function TheoryChapter 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 EquationsChapter 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 EquationsChapter 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 EquationsChapter 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 EquationsChapter 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