Foreword -- 1 Functions of a Complex Variable -- 1.1 Introduction -- 1.2 Complex Numbers -- 1.2.1 Complex Arithmetic -- 1.2.2 Graphical Representation and the Polar Form -- 1.2.3 Curves and Regions in the Complex Plane -- 1.3 Functions of a Complex Variable -- 1.3.1 Basic Concepts -- 1.3.2 Continuity, Differentiability and Analyticity -- 1.4 Power Series -- 1.5 The Elementary Functions -- 1.5.1 Rational Functions -- 1.5.2 The Exponential Function -- 1.5.3 The Trigonometric and Hyperbolic Functions -- 1.5.4 The Logarithm -- 1.5.5 The General Power z -- 1.6 Multivalued Functions and Riemann Surfaces -- 1.7 Conformal Mapping -- 2 Cauchy's Theorem -- 2.1 Complex Integration -- 2.2 Cauchy's Theorem -- 2.2.1 Statement and Proof -- 2.2.2 Path Independence -- 2.2.3 The Fundamental Theorem of Calculus -- 2.3 Further Consequences of Cauchy's Theorem -- 2.3.1 Cauchy's Integral -- 2.3.2 Cauchy's Derivative Formula -- 2.3.3 The Maximum Modulus Principle -- 2.3.4 The Cauchy-Liouville Theorem -- 2.4 Power Series Representations of Analytic Functions -- 2.4.1 Uniform Convergence -- 2.4.2 Taylor's Theorem -- 2.4.3 Laurent's Theorem -- 2.4.4 Practical Methods for Generating Power Series -- 2.5 Zeros and Singularities -- 3 The Calculus of Residues -- 3.1 The Residue Theorem -- 3.2 Calculating Residues -- 3.3 Evaluating Definite Integrals -- 3.3.1 Angular Integrals -- 3.3.2 Improper Integrals of Rational Functions -- 3.3.3 Improper Integrals Involving Trigonometric Functions -- 3.3.4 Improper Integrals Involving Exponential Functions -- 3.3.5 Integrals Involving Many-Valued Functions -- 3.3.6 Deducing Integrals from Others -- 3.3.7 Singularities on the Contour and Principal Value Integrals -- 3.4 Summation of Series -- 3.5 Representation of Meromorphic Functions -- 4 Dispersion Representations -- 4.1 From Cauchy Integral Representation to Hilbert Transforms.

4.2 Adding Poles and Subtractions -- 4.3 Mathematical Applications -- 5 Analytic Continuation -- 5.1 Analytic Continuation -- 5.2 The Gamma Function -- 5.3 Integral Representations and Integral Transforms -- 6 Asymptotic Expansions -- 6.1 Asymptotic Series -- 6.2 Watson's Lemma -- 6.3 The Method of Steepest Descent -- 7 Padé Approximants -- 7.1 From Power Series to Padé Sequences -- 7.2 Numerical Experiments -- 7.3 Stieltjes Functions -- 7.4 Convergence Theorems -- 7.5 Type II Padé Approximants -- 8 Fourier Series and Transforms -- 8.1 Trigonometrical and Fourier Series -- 8.2 The Convergence Question -- 8.3 Functions Having Arbitrary Period -- 8.4 Half-Range Expansions - Fourier Sine and Cosine Series -- 8.5 Complex Form of Fourier Series -- 8.6 Transition to Fourier Transforms -- 8.7 Examples of Fourier Transforms -- 8.8 The Dirac Delta Function and Transforms of Distributions -- 8.9 Properties of Fourier Transforms -- 8.10 Fourier Sine and Cosine Transforms -- 8.11 Laplace Transforms -- 8.12 Application: Solving Differential Equations -- 9 Ordinary Linear Differential Equations -- 9.1 Introduction -- 9.2 Linear DE' s of Second Order -- 9.3 Given One Solution, Find the Others -- 9.4 Finding a Particular Solution for Homogeneous DE's -- 9.5 Method of Frobenius -- 9.6 The Legendre Differential Equation -- 9.7 Bessel's Differential Equation -- 9.8 Some Other Tricks of the Trade -- 9.8.1 Expansion About the Point at Infinity -- 9.8.2 Factorization of the Behaviour at Infinity -- 9.8.3 Changing the Independent Variable -- 9.8.4 Changing the Dependent Variable -- 9.9 Solution by Definite Integrals -- 9.10 The Hypergeometric Equation -- 10 Partial Differential Equations and Boundary Value Problems -- 10.1 The Partial Differential Equations of Mathematical Physics -- 10.2 Curvilinear Coordinates -- 10.3 Separation of Variables.

10.4 What a Difference the Choice of Coordinate System Makes! -- 10.5 The Sturm-Liouville Eigenvalue Problem -- 10.6 A Convenient Notation (And Another Algebraic Digression) -- 10.7 Fourier Series and Transforms as Eigenfunction Expansions -- 10.8 Normal Mode (or Initial Value) Problems -- 11 Special Functions -- 11.1 Introduction -- 11.2 Spherical Harmonics: Problems Possessing Spherical Symmetry -- 11.2.1 Introduction -- 11.2.2 Associated Legendre Polynomials -- 11.2.3 Properties of Legendre Polynomials -- 11.2.4 Problems Possessing Azimuthal Symmetry -- 11.2.5 Properties of the Associated Legendre Polynomials -- 11.2.6 Completeness and the Spherical Harmonics -- 11.2.7 Applications: Problems Without Azimuthal Symmetry -- 11.3 Bessel Functions: Problems Possessing Cylindrical Symmetry -- 11.3.1 Properties of Bessel and Neumann Functions -- 11.3.2 Applications -- 11.3.3 Modified Bessel Functions -- 11.3.4 Electrostatic Potential in and around Cylinders -- 11.3.5 Fourier-Bessel Transforms -- 11.4 Spherical Bessel Functions: Spherical Waves -- 11.4.1 Properties of Spherical Bessel Functions -- 11.4.2 Applications: Spherical Waves -- 11.5 The Classical Orthogonal Polynomials -- 11.5.1 The Polynomials and Their Properties -- 11.5.2 Applications -- 12 Non-Homogeneous Boundary Value Problems: Green's Functions -- 12.1 Ordinary Differential Equations -- 12.1.1 Definition of a Green's Function -- 12.1.2 Direct Construction of the Sturm Liouville Green' s Function -- 12.1.3 Application: The Bowed Stretched String -- 12.1.4 Eigenfunction Expansions : The Bilinear Formula -- 12.1.5 Application: the Infinite Stretched String -- 12.2 Partial Differential Equations -- 12.2.1 Green's Theorem and Its Consequences -- 12.2.2 Poisson's Equation in Two Dimensions and With Rectangular Symmetry -- 12.2.3 Potential Problems in Three Dimensions and the Method of Images.

12.2.4 Expansion of the Dirichlet Green's Function for Poisson's Equation When There Is Spherical Symmetry -- 12.2.5 Applications -- 12.3 The Non-Homogeneous Wave and Diffusion Equations -- 12.3.1 The Non-Homogeneous Helmholtz Equation -- 12.3.2 The Forced Drumhead -- 12.3.3 The Non-Homogeneous Helmholtz Equation With Boundaries at Infinity -- 12.3.4 General Time Dependence -- 12.3.5 The Wave and Diffusion Equation Green's Functions for Boundaries at Infinity -- 13 Integral Equations -- 13.1 Introduction -- 13.2 Types of Integral Equations -- 13.3 Convolution Integral Equations -- 13.4 Integral Equations With Separable Kernels -- 13.5 Solution by Iteration: Neumann Series -- 13.6 Hilbert Schmidt Theory -- Bibliography -- Bibliography.