Trigonometric Series and Fourier Series

Abstract

D. Bernoulli, D’ Alembert, Lagrange, and Euler, from about 1740 onward, were led by problems in mathematical physics to consider and discuss heatedly the possibility of representing a more or less arbitrary function f with period 2n as the sum of a trigonometric series of the form

$$\frac{1}{2} {{a}_{0}} + \sum\limits_{{n = 1}}^{{}} {\left( {{{a}_{n}}\cos nx + {{b}_{n}}\sin nx} \right),}$$

or of the formally equivalent series in its so-called “complex” form

$$\sum\limits_{n = - \infty }^\infty {{c_n}{e^{inx}},} $$

in which, on writing b₀ = 0, the coefficients c_n are given by the formulae

$$\frac{1}{2} \left( {{{a}_{n}} - i{{b}_{n}}} \right),{{c}_{{ - n}}} = \frac{1}{2} \left( {{{a}_{n}} + i{{b}_{n}}} \right){\text{ }}\left( {n = 0,1,2, \ldots } \right).$$

This discussion sparked off one of the crises in the development of analysis.