ASN6c - Fourier Transform property: Value at origin

Value at Origin

This Fourier Transform property says $s(0)$ transforms to $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)df$

$s(0)=S(f){|}_{f=0}=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)e^{-j2\pi ft}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt$

And $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt$ in time transforms in frequency to $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)df$

The result is $s(0)$ transforms to $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)df$