Fourier series

**Ulysses** · May 18th 2011, 02:57 PM

Hi there. I have to find the Fourier development for $f(t)=\begin{Bmatrix}{1 }&\mbox{ if }& 0<t<1\\0 & \mbox{if}& 1<t<2\end{matrix}$

So this is what I did, I think its wrong, but I don't know where I did the mistake.
$a_0=\displaystyle\frac{1}{4}\displaystyle\int_{0}^ {1} dt=1/4$

$a_n=\displaystyle\frac{1}{2}\displaystyle\int_{0}^ {1}\cos \left( \displaystyle\frac{n\pi t}{2} \right) dt=\displaystyle\frac{1}{n\pi}\sin \left( \displaystyle\frac{n\pi}{2}\right)$

$b_n=\displaystyle\frac{1}{2}\displaystyle\int_{0}^ {1}\sin \left( \displaystyle\frac{n\pi t}{2} \right) dt=\displaystyle\frac{1}{2}\left[\displaystyle\frac{-2}{n\pi}\cos \left( \displaystyle\frac{n\pi t}{2} \right) \right]_0^1=\displaystyle\frac{-1}{n\pi}\cos \left( \displaystyle\frac{n\pi}{2} \right)+\displaystyle\frac{1}{n\pi}$

Then:
$f(t)\sim{ \displaystyle\frac{1}{4}+ \displaystyle\sum_{n=1}^{\infty} \displaystyle\frac{1}{n\pi} \sin \left( \displaystyle\frac{n\pi}{2}\right) - \displaystyle\sum_{n=1}^{\infty} \left[ \displaystyle\frac{1}{n\pi}\cos\left( \displaystyle\frac{n\pi}{2}\right)-\displaystyle\frac{1}{n\pi}\right]}$

When I plot this on mathematica I get a line. So, I think I've made a mistake somewhere, but I don't know where the error is. Somehow I have to define the zero intervals, I think I should use the heaviside function.

Bye and thanks for 'ur help.

PS:I didn't know where to post this, so move it if it should be in another subforum.

**TKHunny** · May 18th 2011, 05:59 PM

You probably should have a 't' in there, somewhere. It is f(t), after all. You did calculate coefficients. Coefficients on what?

**Ulysses** · May 18th 2011, 06:31 PM

You're right. I'll correct this later, I'm a bit erratic right now :P

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