So I'm trying to find the Fourier series of the $PC_{2\pi}$ function $e^{-|x|}$ for $x\in[-\pi,\pi]$ using the formula
$$c_0+\sum_{k=-\infty}^\infty c_k e^{-ikx}$$
where
$$ c_k(f)=\frac{1}{2\pi} \int_{-\pi}^\pi f(x)e^{-ikx} dx $$ I've reached an answer (below), but I think that the result is wrong. Is my answer correct? If not, then what is the correct answer?
In this case that would be finding
$$c_k(f)=\frac{1}{2\pi} \int_{-\pi}^\pi e^{-|x|}e^{-ikx} dx$$
which I've separated into
$$=\frac{1}{2\pi} \int_{-\pi}^0 e^{x-ikx} dx + \int_0^\pi e^{-x-ikx}dx $$
For each of them I found
\begin{align} \int_{-\pi}^{0} e^{x-i k x} d x &= \int e^{t} \cdot \frac{1}{1-i k} d t \\ &= \int \frac{e^{t}(1+i k)}{1+k^{2}} d t \\ &= \int \frac{e^{t}}{1+k^{2}}+i \frac{k e^{t}}{1+k^{2}} d t \\ &= \int_{-\pi}^{0} \frac{e^{x-i k x}}{1+k^{2}} d t+\int_{-\pi}^{0} i \frac{k e^{x-i k x}}{1+k^{2}} d t \\ &=\frac{1-e^{-\pi+i \pi k}}{1+k^{2}}+\frac{i k}{1+k^{2}}\left(1-e^{-\pi+i \pi k}\right) \end{align}
and
\begin{align} \int_{0}^{\pi} e^{-x-i k x} d x &= \int e^{t} \cdot \frac{1}{-1-i k} d t \\ &= \int \frac{e^{t}(-1+i k)}{1+k^{2}} d t \\ &= \int-\frac{e^{t}}{1+k^{2}}+i \frac{k e^{t}}{1+k^{2}} d t \\ &= \int_{0}^{\pi}-\frac{e^{-x-i k x}}{1+k^{2}} d t+\int_{-\pi}^{0} i \frac{k e^{-x-i k x}}{1+k^{2}} d t \\ &=\frac{e^{-\pi-i \pi k}-1}{1+k^{2}}+i \frac{k}{1+k^{2}}\left(e^{-\pi-i \pi k}-1\right) \end{align}
Hence my final result: $$ c_k(f)= \frac{e^{i\pi k}+e^{-\pi-i\pi k}+ike^{-\pi-i\pi k}-ike^{i\pi k-\pi}}{2\pi(1+k^2)}$$
The Fourier cosine series for $e^{-\left|x\right|}$ is as follows.
(1) $\quad e^{-\left|x\right|}=\frac{1+\sinh (\pi )-\cosh(\pi)}{\pi}+\frac{2}{\pi}\underset{K\to\infty}{\text{lim}}\ \sum\limits_{k=1}^K \frac{\left(1-(-1)^k e^{-\pi }\right)}{k^2+1}\cos(k\ x)$
Figure (1) below illustrates formula (1) above for $e^{-\left|x\right|}$ evaluated at $K=100$ in orange overlaid on the blue reference function.
Figure (1): Illustration of formula (1) for $e^{-\left|x\right|}$
The following links provide information on the derivation of Fourier cosine series for even functions.
Wikipedia: Cosine series
Weisstein, Eric W. "Fourier Cosine Series." From MathWorld--A Wolfram Web Resource.