Heaviside Step Function - Fourier Transform

Fourier Transform

The Fourier transform of the Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have

$\hat{H}(s) = \lim_{N\to\infty}\int^N_{-N} \mathrm{e}^{-2\pi i x s} H(x)\,{\rm d}x = \frac{1}{2} \left( \delta(s) - \frac{i}{\pi}\text{p.v.}\frac{1}{s} \right).$

Here is the distribution that takes a test function to the Cauchy principal value of The limit appearing in the integral is also taken in the sense of (tempered) distributions.

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