I want to determine the fourier coefficients and series of $$ f(x) = e^{z|x|} $$ where $ z\in \mathbb{C} \backslash \{0 \} $ on the intervall $ [0, 2\pi ]$.
So let's get started:
$ \hat{f} (n)= \frac{1}{2 \pi} \int_0^{2 \pi} e^{z|x|} e^{-inx} dx $
$ = \frac{1}{2 \pi}\int_0^{2 \pi} 1e^{zx-inx} dx $
=$ \frac{1}{2 \pi} [[ \frac{1}{z-in} e^{zx-inx} ]_0^{2 \pi} - \int_0^{2\pi} 0 dx ]$ (partial integration)
=$\frac{1}{2 \pi} [ \frac{1}{z-in} ( e^{2\pi z} e^{-2 \pi i z} -1 )]$
I am new to fourieranalysis, so I am not sure if thats on the right way.. also not sure how to proceed or doing it another way
thanks for any help!