## Fourier Transformation of the Heaviside Function

The Heaviside step function is very important in physics. It often models a sudden switch-on phenomenon and is therefore present in a lot of integrals. For example, the derivation of the Kramers-Kronig Relations can be significantly simplified once we know the Fourier-Transform $$\bar{\theta}(\omega)$$ of the Heaviside function $$\theta(t)$$. Although the function appears to be quite simple, the calculation of its Fourier transform can be quite challenging. Let's find out!

## Problem Statement

Show that the Fourier transform is given by the following expression:

$2\pi\bar{\theta}(\omega)=\int\limits _{-\infty}^{\infty}\theta(t)e^{i\omega t}dt=P\frac{i}{\omega}+\pi\delta(\omega),$

where $$\delta(\omega)$$ is the Dirac delta distribution and the functional $$P$$ refers to the Cauchy Principal value.