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

Here, ( e^-\alpha t ) ensures convergence for ( \alpha > 0 ). Then:

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

Fourier Transform Step Function Link Info

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

Here, ( e^-\alpha t ) ensures convergence for ( \alpha > 0 ). Then: 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 =