Fourier Series and Boundary Value Problems

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Title
Contents
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 aWave 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 Unbounded 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
	Orthogonal Eigenfunctions and 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 Jν (x)
	Solutions When ν = 0, 1, 2,
	Recurrence Relations
	Bessel’s Integral Form
	Some Consequences of the Integral Forms
	The Zeros of Jn(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 Pn(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 theWave Equation
Appendixes
Bibliography
Some Fourier Series Expansions
Solutions of Some Regular Sturm-Liouville Problems
Some Fourier-Bessel Series Expansions
Index