# Book:Arne Broman/Introduction to Partial Differential Equations

## Arne Broman: Introduction to Partial Differential Equations from Fourier Series to Boundary-value Problems

Published $\text {1970}$, Dover Publications, Inc.

ISBN 0-486-66158-X.

### Contents

Preface (Göteborg, March 1970)
Chapter $1$ Fourier series
$1.1$ Basic concepts
$1.2$ Fourier series and Fourier coefficients
$1.3$ A minimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem
$1.4$ Convergence of the Fourier series
$1.5$ The Parseval formula
$1.6$ Determination of the sum of certain trigonometric series
Chapter $2$ Orthogonal systems
$2.1$ Integration of complex-valued functions of a real variable
$2.2$ Orthogonal systems
$2.3$ Complete orthogonal systems
$2.4$ Integration of Fourier series
$2.5$ The Gram-Schmidt orthogonalization process
$2.6$ Sturm-Liouville problems
Chapter $3$ Orthogonal polynomials
$3.1$ The Legendre polynomials
$3.2$ Legendre series
$3.3$ The Legendre differential equation. The generating function of the Legendre polyomials
$3.4$ The Tchebycheff polynomials
$3.5$ Tchebycheff series
$3.6$ The Hermite polynomials. The Laguerre polynomials
Chapter $4$ Fourier transforms
$4.1$ Infinite interval of integration
$4.2$ The Fourier integral formula: a heuristic introduction
$4.3$ Auxiliary theorems
$4.4$ Proof of the Fourier integral formula. Fourier transforms
$4.5$ The convolution theorem. The Parseval formula
Chapter $5$ Laplace transforms
$5.1$ Definition of the Laplace transform. Domain. Analyticity
$5.2$ Inversion formula
$5.3$ Further properties of Laplace transforms. The convolution theorem
$5.4$ Applications to ordinary differential equations
Chapter $6$ Bessel functions
$6.1$ The gamma function
$6.2$ The Bessel differential equation. Bessel functions
$6.3$ Some particular Bessel functions
$6.4$ Recursion formulas for the Bessel functions
$6.5$ Estimation of Bessel functions for large values of $x$. The zeros of the Bessel functions
$6.6$ Bessel series
$6.7$ The generating function of the Bessel functions of integral order
$6.8$ Neumann function
Chapter $7$ Partial differential equations of first order
$7.1$ Introduction
$7.2$ The differential equation of a family of surfaces
$7.3$ Homogeneous differential equations
$7.4$ Linear and quasilinear differential equations
Chapter $8$ Partial differential equations of second order
$8.1$ Problems in physics leading to partial differential equations
$8.2$ Definitions
$8.3$ The wave equation
$8.4$ The heat equation
$8.5$ The Laplace equation