ASN6c - Fourier Transform property: Value at origin

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Value at Origin

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

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

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

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