1. ## fourier integral

I'm having problems with understanding Fourier integral. I only have one example solved, but i don't have a clue what is done there.

I have to transform this function into a Fourier integral

$f(x) = \left\{
\begin{array}{c l}
A\cos{\omega_0 x},x\epsilon <-\frac{\pi}{2\omega_0},\frac{\pi}{2\omega_0}> \\
0 ,\mbox{ otherwise}
\end{array}\right.
$

What next?

2. Originally Posted by Pinsky
I'm having problems with understanding Fourier integral. I only have one example solved, but i don't have a clue what is done there.

I have to transform this function into a Fourier integral

$f(x) = \left\{
\begin{array}{c l}
A\cos{\omega_0 x},x\epsilon <-\frac{\pi}{2\omega_0},\frac{\pi}{2\omega_0}> \\
0 ,\mbox{ otherwise}
\end{array}\right.
$

What next?

$\mathcal{F}f (\omega) = \int_{-\infty}^{\infty} f(x) e^{i\omega x}dx$

(your definition may vary as there is no standard about where the constants go in the FT IFT pair).

Now your function is zero outside of the inteval $\left[-\frac{\pi}{2\omega_0},\frac{\pi}{2\omega_0}\right]$, so the integral can be done over this interval:

$\mathcal{F}f (\omega) = \int_{-\frac{\pi}{2\omega_0}}^{\frac{\pi}{2\omega_0}} f(x) e^{i\omega x}dx=
\int_{-\frac{\pi}{2\omega_0}}^{\frac{\pi}{2\omega_0}} A\cos(\omega_0 x)\;e^{i\omega x}dx$

and I will leave the rest to you.

RonL