Fourier Integrals

Abstract

We have seen that periodic functions can be expanded in Fourier series. Recall that, if the fundamental interval of the function $f(x)$ is $(a,\, b)$ so that the period is $L = (b-a)$, the expansion and inversion formulas (Eqs. (17.4) and (17.5)) are

$$\begin{aligned} f(x) = (1/L)\sum _{n =-\infty }^{\infty }f_{n}\, e^{2\pi nix/L}\quad \text {and} \quad f_{n} = \int _{a}^{b}dx\, f(x)\, e^{-2\pi nix/L}. \end{aligned}$$

What happens if the function $f(x)$ is not periodic?