4 - Fourier Transform

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