Fourier transform inversion using an elementary differential equation and a contour integral

Abstract

Let $f$ be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that $f$ is absolutely continuous such that $f$ and $f'$ are Lebesgue integrable. A function $g$ is defined by $f'(t)-iwf(t)=g(t)$. This differential equation has a well known integral solution using the Heaviside step function. An elementary calculation with residues is used to write the Heaviside step function as a simple contour integral. The rest of the proof requires elementary manipulation of integrals. Hence, the Fourier transform inversion theorem is proved with very little machinery. With only minor changes the method is also used to prove the inversion theorem for functions of several variables and to prove Riemann's localization theorem.