ASN6c - Fourier Transform property: Value at origin

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Value at Origin

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

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

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

The result is $s(0)$ transforms to $\int _{-\infty }^{\infty }S(f)df$