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Summary
Published by McGraw-Hill since its first edition in 1941, this classic text is an introduction to Fourier series and their applications to boundary value problems in partial differential equations of engineering and physics. It will primarily be used by students with a background in ordinary differential equations and advanced calculus. There are two main objectives of this text. The first is to introduce the concept of orthogonal sets of functions and representations of arbitrary functions in series of functions from such sets. The second is a clear presentation of the classical method of separation of variables used in solving boundary value problems with the aid of those representations. The book is a thorough revision of the seventh edition and much care is taken to give the student fewer distractions when determining solutions of eigenvalue problems, and other topics have been presented in their own sections like Gibbs' Phenomenon and the Poisson integral formula.
Table of Contents
Preface
1 Fourier Series
Piecewise Continuous Functions
Fourier Cosine Series
Examples
Fourier Sine Series
Examples
Fourier Series
Examples
Adaptations to Other Intervals
2 Convergence of Fourier Series
One-Sided Derivatives
A Property of Fourier Coefficients
Two Lemmas
A Fourier Theorem
A Related Fourier Theorem
Examples
Convergence on Other Intervals
A Lemma
Absolute and Uniform Convergence of Fourier Series
The Gibbs Phenomenon
Differentiation of Fourier Series
Integration of Fourier Series
3 Partial Differential Equations of Physics
Linear Boundary Value Problems
One-Dimensional Heat Equation
Related Equations
Laplacian in Cylindrical and Spherical Coordinates
Derivations
Boundary Conditions
Duhamel's Principle
A Vibrating String
Vibrations of Bars and Membranes
General Solution of the Wave Equation
Types of Equations and Boundary Conditions
4 The Fourier Method
Linear Operators
Principle of Superposition
Examples
Eigenvalues and Eigenfunctions
A Temperature Problem
A Vibrating String Problem
Historical Development
5 Boundary Value Problems
A Slab with Faces at Prescribed Temperatures
Related Temperature Problems
Temperatures in a Sphere
A Slab with Internally Generated Heat
Steady Temperatures in Rectangular Coordinates
Steady Temperatures in Cylindrical Coordinates
A String with Prescribed Initial Conditions
Resonance
An Elastic Bar
Double Fourier Series
Periodic Boundary Conditions
6 Fourier Integrals and Applications
The Fourier Integral Formula
Dirichlet's Integral
Two Lemmas
A Fourier Integral Theorem
The Cosine and Sine Integrals
Some Eigenvalue Problems on Undounded Intervals
More on Superposition of Solutions
Steady Temperatures in a Semi-Infinite Strip
Temperatures in a Semi-Infinite Solid
Temperatures in an Unlimited Medium
7 Orthonormal Sets
Inner Products and Orthonormal Sets
Examples
Generalized Fourier Series
Examples
Best Approximation in the Mean
Bessel's Inequality and Parseval's Equation
Applications to Fourier Series
8 Sturm-Liouville Problems and Applications
Regular Sturm-Liouville Problems
Modifications
Orthogonality of Eigenfunctions adn Real Eigenvalues
Real-Valued Eigenfunctions
Nonnegative Eigenvalues
Methods of Solution
Examples of Eigenfunction Expansions
A Temperature Problem in Rectangular Coordinates
Steady Temperatures
Other Coordinates
A Modification of the Method
Another Modification
A Vertically Hung Elastic Bar
9 Bessel Functions and Applications
The Gamma Function
Bessel Functions J_{n}(x)
Solutions When v = 0,1,2,...
Recurrence Relations
Bessel's Integral Form
Some Consequences of the Integral Forms
The Zeros of J_{n}(x)
Zeros of Related Functions
Orthogonal Sets of Bessel Functions
Proof of the Theorems
Two Lemmas
Fourier-Bessel Series
Examples
Temperatures in a Long Cylinder
A Temperature Problem in Shrunken Fittings
Internally Generated Heat
Temperatures in a Long Cylindrical Wedge
Vibration of a Circular Membrane
10 Legendre Polynomials and Applications
Solutions of Legendre's Equation
Legendre Polynomials
Rodrigues' Formula
Laplace's Integral Form
Some Consequences of the Integral Form
Orthogonality of Legendre Polynomials
Normalized Legendre Polynomials
Legendre Series
The Eigenfunctions P_{n}(cos θ)
Dirichlet Problems in Spherical Regions
Steady Temperatures in a Hemisphere
11 Verification of Solutions and Uniqueness
Abel's Test for Uniform Convergence
Verification of Solution of Temperature Problem
Uniqueness of Solutions of the Heat Equation
Verification of Solution of Vibrating String Problem
Uniqueness of Solutions of the Wave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems