Fourier series
Basic concepts
Fourier series and Fourier coefficients
A mimimizing property of the Fourier coefficients
The Riemann-Lebesgue theorem
Convergence of Fourier series
The Parseval formula
Determination of the sum of certain trigonemetric series
Orthogonal systems
Integration of complex-valued functions of a real variable
Orthogonal systems
Complete orthogonal systems
Integration of Fourier series
The Gram-Schmidt orthogonalization process
Sturm-Liouville problems
Orthogonal polynomials
The Legendre polynomials
Legendre series
The Legendre differential equation
The generating function of the Legendre polynomials
The Tchebycheff polynomials
Tchebycheff series
The Hermite polynomials
The Laguerre polynomials
Fourier transforms
Infinite interval of integration
The Fourier integral formula: a heuristic introduction
Auxiliary theorems
Proof of the Fourier integral formula
Fourier transforms
The convention theorem
The Parseval formula
Laplace transforms
Definition of the Laplace transform
Domain
Analyticity
Inversion formula
Further properties of Laplace transforms
The convolution theorem
Applications to ordinary differential equations
Bessel functions
The gamma function
The Bessel differential equation
Bessel functions
Some particular Bessel functions
Recursion formulas for the Bessel functions
Estimation of Bessel functions for large values of x
The zeros of the Bessel functions
Bessel series
The generating function of the Bessel functions of integral order
Neumann functions
Partial differential equations of first order
Introduction
The differential equation of a family of surfaces
Homogeneous differential equations
Linear and quasilinear differential equations
Partial differential equations of second order
Problems in physics leading to partial differential equations
Definitions
The wave equation
The heat equation
The Laplace equation
Answers to exercises
Bibliography
Conventions
Symbols
Index
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