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

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

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

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

What next?

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

(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 $\displaystyle \left[-\frac{\pi}{2\omega_0},\frac{\pi}{2\omega_0}\right]$, so the integral can be done over this interval:

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

and I will leave the rest to you.

RonL