Brief Historical Introduction

Abstract

Historically, Joseph Fourier (1770–1830) first introduced the remarkable idea of expansion of a function in terms of trigonometric series without giving any attention to rigorous mathematical analysis. The integral formulas for the coefficients of the Fourier expansion were already known to Leonardo Euler (1707–1783) and others. In fact, Fourier developed his new idea for finding the solution of heat (or Fourier) equation in terms of Fourier series so that the Fourier series can be used as a practical tool for determining the Fourier series solution of partial differential equations under prescribed boundary conditions. Thus, the Fourier series of a function f(x) defined on the interval (-ℓ, ℓ) is given by

$$ f\left( x \right) = \sum\limits_{n = - \infty }^\infty {c_n } {\rm{exp}}\left( {\frac{{in\pi x}}{\ell }} \right), $$

(1.1.1)

where the Fourier coefficients are

$$ c_n = \frac{{\rm{1}}}{{{\rm{2}}\ell }}\int\limits_{ - \ell }^\ell {f\left( t \right)} {\rm{exp}}\left( { - \frac{{in\pi t}}{\ell }} \right)dt. $$

(1.1.2)