1. ## Fourier Series Help

So I found the complex version of the fourier series of e^x.

$\sum_{n=-\infty}^{\infty}(-1)^n(\frac{l+in\pi}{l^2+n^2{\pi}^2})(sinh(l))e^{\f rac{in{\pi}x}{l}}$

I was wondering if there is a way to convert this to the real full fourier series form.

Thanks

2. (Pretend all the following functions are well-behaved).

Let $f(x) = \sum_{n=-\infty}^{\infty} a_n e^{inx}$ where $f(x)$ is a $2\pi$-periodic function.
We can write,
$f(x) = a_0 + \sum_{n=1}^{\infty} [(a_n+a_{-n})\cos nt + i (a_n + a_{-n}) \sin nt ]$