出版说明
序
1 Overture
1.1 Some equations of mathematical physics
1.2 Linear differential operators
1.3 Separation of variables
2 Fourier Series
2.1 The Fourier series of a periodic function
2.2 A convergence theorem
2.3 Derivatives, integrals, and uniform convergence
2.4 Fourier series on intervals
2.5 Some applications
2.6 Further remarks on Fourier series
3 Orthogonal Sets of Functions
3.1 Vectors and inner products
3.2 Functions and inner products
3.3 Convergence and completeness
3.4 More about L2 spaces; the dominated convergence theorem
3.5 Regular Sturm-Liouville problems
3.6 Singular Sturm-Liouville problems
4 Some Boundary Value Problems
4.1 Some useful techniques
4.2 One-dimensional heat flow
4.3 One-dimensional wave motion
4.4 The Dirichlet problem
4.5 Multiple Fourier series and applications
5 Bessel Functions
5.1 Solutions of Bessel#s equation
5#.2 Bessel function identities
5.3 Asymptotics and zeros of Bessel functions
5.4 Orthogonal sets of Bessel functions
5.5 Applications of Bessel functions
5.6 Variants of Bessel functions
6 Orthogonal Polynomials
6.1 Introduction
6.2 Legendre polynomials
6.3 Spherical coordinates and Legendre functions
6.4 Hermite polynomials
6.5 Laguerre polynomials
6.6 Other orthogonal bases
7 The Fourier Transform
7.1 Convolutions
7.2 The Fourier transform
7.3 Some applications
7.4 Fourier transforms and Sturm-Liouville problems
7.5 Multivariable convolutions and Fourier transforms
7.6 Transforms related to the Fourier transform
8 The Laplace Transform
8.1 The Laplace transform
8.2 The inversion formula
8.3 Applications: Ordinary differential equations
8.4 Applications: Partial differential equations
8.5 Applications: Integral equations
8.6 Asymptotics of Laplace transforms
9 Generalized Functions
9.1 Distributions
9.2 Convergence, convolution, and approximation
9.3 More examples: Periodic distributions and finite parts
9.4 Tempered distributions and Fourier transforms
9.5 Weak solutions of differential equations
10 Green#s Functions
10.1 Green#s functions for ordinary differential operators
10.2 Green#s functions for partial differential operators
10.3 Green#s functions and regular Sturm-Liouville problems
10.4 Green#s functions and singular Sturm-Liouville problems
Appendices
1 Some physical derivations
2 Summary of complex variable theory
3 The gamma function
4 Calculations in polar coordinates
5 The fundamental theorem of ordinary differential equations
Answers to the Exercises
References
Index of Symbols
Index