Fourier Transform Step Function -

[ \mathcalFu(t) = \frac12 \cdot 2\pi\delta(\omega) + \frac12 \cdot \frac2i\omega = \pi\delta(\omega) + \frac1i\omega ]

[ \int_0^\infty e^-\alpha t e^-i\omega t dt = \int_0^\infty e^-(\alpha + i\omega) t dt = \frac1\alpha + i\omega ] fourier transform step function

[ F(\omega) = \int_-\infty^\infty f(t) e^-i\omega t dt ] [ \mathcalFu(t) = \frac12 \cdot 2\pi\delta(\omega) + \frac12

At first glance, finding its Fourier transform seems impossible. The Fourier transform of a function ( f(t) ) is: fourier transform step function