Green's Functions with Applications, 2nd Edition

Chapter 2

Background Material

One of the fundamental problems of field theory1 is the construction of solutions to linear differential equations when there is a specified source and the differential equation must satisfy certain boundary conditions. The purpose of this book is to show how Green’s functions provide a method for obtaining these solutions. In this chapter, some of the mathematical aspects necessary for developing this technique are presented.

2.1 Fourier Transform

The Fourier transform is the natural extension of Fourier series to a function f (t) of infinite period and is defined by the pair of integrals:

$\begin{array}{c}f\left(t\right)=\frac{1}{2\pi }{\displaystyle {\int }_{-\infty }^{\infty }F\left(\omega \right)}{e}^i\end{array}$