ASN6c - Fourier Transform property: Value at origin

Value at Origin

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

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

Thefore $s(0)=\int _{-\infty }^{\infty }s(t)dt$ in time transformed to frequency is $\int _{-\infty }^{\infty }S(f)df$