Fourier serie find $c_n$

Question

We let $f\in$$PC_{2\pi}$ and $f$ be $f(x)=e^{-|x|}$, $x\in[-\pi,\pi$]. I have to find the Fourier serie for $f$. I have found the Fourier coefficients: $$c_0= \frac{1}{2\pi}\int_{-\pi}^\pi e^{-|x|}dx=\frac{(-1+e^{\pi})e^{-\pi}}{\pi}$$ because $$\int_{}^{}e^{-|x|}dx = \begin{cases} e^{x}, & x \leq 0\\ -e^{-x}+2, & 0<x \\ \end{cases} $$ and I have found with Wolfram Alpha: $$c_n= \frac{1}{2\pi}\int_{-\pi}^\pi e^{-|x|}e^{-ikx} dx=\frac{e^{\pi(-1-ik)}(e^{2i\pi k}(-1-ik)+2e^{\pi+I\pi k}+ik-1)}{2\pi(k^2+1)}$$. I think the calculation for $c_0$ is okay, but how can I formally show the calculation $c_n$? I need some help?

Answer 1

Hint: write the integral as follows: $$ \int_{- \pi}^{0} e^{x-ikx} dx + \int_{0}^{\pi} e^{-x-ikx} dx$$ This way, you avoid the use if the absolute value. Now, simple calculations of the above integrals will give you (after multiplying by $\frac{1}{2 \pi} $) the result.