导语

  

内容提要

  

    本书介绍了傅里叶级数及其在工程和物理学偏微分方程边界值问题中的应用。本书是为已经完成了常微分方程第一课的学生设计的，证明了高等微积分中偶尔需要的更微妙的结果。

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 a 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 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
  ATemperature 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,(x)
  Solutions Whenv
  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
  Iwo 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 Pa(cose)
  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
  Some Fourier-Bessel Series Expansions
Index
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