Mathematical Methods in Physics: Partial Differential Equations, Fourier Series, and Special Functions

 Content: Fourier Series  Periodic Processes and Periodic Functions  Fourier Formulas  Orthogonal Systems of Functions  Convergence of Fourier Series  Fourier Series for Non-periodic Functions  Fourier Expansions on Intervals of Arbitrary Length  Fourier Series in Cosine or Sine Functions  The Complex Form of the Fourier Series  Complex Generalized Fourier Series  Fourier Series for Functions of Several Variables  Uniform Convergence of Fourier Series  The Gibbs Phenomenon  Completeness of a System of Trigonometric Functions  General Systems of Functions: Parseval\'s Equality and Completeness  Approximation of Functions in the Mean  Fourier Series of Functions Given at Discrete Points  Solution of Differential Equations by Using Fourier Series  Fourier Transforms  The Fourier Integral  Problems  Sturm-Liouville Theory  The Sturm-Liouville Problem  Mixed Boundary Conditions  Examples of Sturm-Liouville Problems  Problems  One-Dimensional Hyperbolic Equations  Derivation of the Basic Equations  Boundary and Initial Conditions  Other Boundary Value Problems: Longitudinal Vibrations of a Thin Rod  Torsional Oscillations of an Elastic Cylinder  Acoustic Waves  Waves in a Shallow Channel  Electrical Oscillations in a Circuit  Traveling Waves: D\'Alembert Method  Semi-infinite String Oscillations and the Use of Symmetry Properties  Finite Intervals: The Fourier Method for One-Dimensional Wave Equations  Generalized Fourier Solutions  Energy of the String  Problems  Two-Dimensional Hyperbolic Equations  Derivation of the Equations of Motion  Oscillations of a Rectangular Membrane  The FourierMethod Applied to Small Transverse Oscillations of a Circular Membrane  Problems  One-Dimensional Parabolic Equations  Physical Problems Described by Parabolic Equations: Boundary Value Problems  The Principle of the Maximum, Correctness, and the Generalized Solution  The Fourier Method of Separation of Variables for the Heat Conduction Equation  Heat Conduction in an Infinite Bar  Heat Equation for a Semi-infinite Bar  Problems  Parabolic Equations for Higher-Dimensional Problems  Heat Conduction in More than One Dimension  Heat Conduction within a Finite Rectangular Domain  Heat Conduction within a Circular Domain  Problems  Elliptic Equations  Elliptic Partial Differential Equations and Related Physical Problems  The Dirichlet Boundary Value Problem for Laplace\'s Equation in a Rectangular Domain  Laplace\'s and Poisson\'s Equations for Two-Dimensional Domains with Circular Symmetry  Laplace\'s Equation in Cylindrical Coordinates  Problems  Bessel Functions  Boundary Value Problems Leading to Bessel Functions  Bessel Functions of the First Kind  Properties of Bessel Functions of the First Kind: Jn (x)  Bessel Functions of the Second Kind  Bessel Functions of the Third Kind  Modified Bessel Functions  The Effect of Boundaries on Bessel Functions  Orthogonality and Normalization of Bessel Functions  The Fourier-Bessel Series  Further Examples of Fourier-Bessel Series Expansions  Spherical Bessel Functions  The Gamma Function  Problems  Legendre Functions  Boundary Value Problems Leading to Legendre Polynomials  Generating Function for Legendre Polynomials  Recurrence Relations  Orthogonality of Legendre Polynomials  The Multipole Expansion in Electrostatics  Associated Legendre Functions P m (x)   n Orthogonality and the Normof Associated Legendre Functions  Fourier-Legendre Series in Legendre Polynomials  Fourier-Legendre Series in Associated Legendre Functions  Laplace\'s Equation in Spherical Coordinates and Spherical Functions  Problems  Eigenvalues and Eigenfunctions of the Sturm-Liouville Problem Auxiliary Functions for Different Types of Boundary Conditions The Sturm-Liouville Problem and the Laplace Equation  Vector Calculus  How to Use the Software Associated with this Book  Program Overview  Examples Using the Program TrigSeries  Examples Using the Program Waves  Examples Using the Program Heat  Examples Using the Program Laplace  Examples Using the Program FourierSeries