Originally Posted by

**liquidFuzz** I'm trying to brush the dust of my transform tinkering. (Have to use it in some electronic stuff.) I just did an exercise where I calculate the F-transform of an even function.

Let f(x) be:

$\displaystyle \displaystyle f(t)=\left\{\begin{matrix} 0 & x > 2\\ 2-x & 0 < x < 2 \end{matrix}\right.$

Find the even Fourier transform.

I calculated the transform with this formula

$\displaystyle \displaystyle f(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt = \int_{-\infty}^{\infty} f(t) (cos(\omega t) + i sin(\omega t))dt$

Knowing that the function is even I write:

$\displaystyle \displaystyle f(\omega) = 2 \int_{0}^{\infty} (2-t) cos(\omega t) dt$

A bit integration gives, don't hesitate to call out errors.

$\displaystyle \displaystyle f(\omega) = \frac{4sin^{2}w}{w^2}$

Now as an application to this calculation I'm supposed to determine this integral.

$\displaystyle \displaystyle I = \int_0^{\infty} \frac{sin^{2}w}{w^2}$

This is where I like to get some help... I simply don't know how to do it. Calculating the integral, in an other way, I get $\displaystyle \displaystyle \frac{\pi}{2}$