I need to compute the fourier series of $f(t)=e^{\cos(t)}, 0 \leq t < 2\pi$. 


The fourier series are defined as
$f(t) = \sum\_{n=-\infty}^\infty c\_n e^{2\pi int/T}$ with $c\_n = \frac 1 T \int\_0^T e^{-2\pi int/T}f(t) \, dt$.


I have tried to do this by using the definition of the $c\_n$, but i get stuck when with the integration by parts. I also have tried to use the approach used in of [this question](https://math.stackexchange.com/questions/97558/prove-fourier-series-of-e-cos-x-sin-sin-x-is-sum-n-0-infty-frac) but i cannot go forward more than expanding $e^{\cos(t)}$.
Could any of you help me with a hint?



Fourier series of $e^{\cos x}$
$$\int\_0^{2\pi} \exp(int) \exp(\cos(t))\; dt = \int\_{-\pi}^{\pi} \cos(n t) \exp(\cos(t))\; dt = 2 \pi I\_n(1)$$ where $I\_n$ is a [modified Bessel function of the first kind](http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html) and thus
$$I\_n(1)=\frac12\sum\_{k\geq0}\frac1{4^kk!(n+k)!}.$$


