4 - Fourier Transform

Kevin Starkey
Find ${ℱ}[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt]$
First we know that ${ℱ}[\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt]=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt)e^{j2\pi ft}dt$
We also know that ${ℱ}[s(t)]=S(f)and\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{j2\pi ft}dt=\delta (f)$
Which gives us $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}(\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}s(t)dt)e^{j2\pi ft}dt=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)\delta (f)df$
Since $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (f)df$ is only non-zero at f = 0 this yeilds
$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}S(f)\delta (f)df=S(0)$
So <math> \mathcal{F}\left[\int_{-\infty}^ \infty s(t)dt\right] = S(0)