Originally Posted by

**StefanM** I'm trying to find the complex form for this fourier series and then transform it into the real one.I don't know exactly how this transformation is suppose to happen..I'm thinking it must have something to do with $\displaystyle e^{-ikt}=\cos (kt)+i\sin (kt)$.Also when I try to calculate the complex form I get zero which means I"m doing something wrong.

F(t) is defined as:

$\displaystyle f(t)=\begin{cases}-1 & ( -\pi\leqslant t<0), \\ 1&(0<t<\pi).\end{cases}$

Now $\displaystyle c_{k}=\int_{-\pi}^\pi f(t) e^{-ikt}dt$

$\displaystyle \Rightarrow\;c_{k}=\frac{1}{2\pi}\Bigl(\int_{-\pi}^0 -e^{-ikt}dt+\int_0^\pi e^{-ikt}dt\Bigr) $

. . . .. . .$\displaystyle = \frac{1}{2\pi ik}\Bigl[1-e^{-ikt}+e^{ikt}-1\Bigr]=0.$

This is not the correct answer.Can someone please help me?