How to integrate $e^{r\cos x} \cos(r\sin x)$

Question

The title says everything. I'm studying fourier series and I've stumbled upon this question:

find the fourier series of $f(x) = e^{r\cos x} \cos(r\sin x)$. So that i need to integrate this function from $-\pi$ to $\pi$

I've tried integration by parts and a few u substitutions and got nowhere.

Answer 1

Hint: first note that $f(x)$ is the real part of $e^{r \cos x} e^{i r \sin x} = e^{r e^{ix}}$. Expand the "outer" exponential in a series...

Answer 2

HINT

Look up Bessel functions. We have $$J_r(x) = \dfrac1{2\pi} \int_{-\pi}^{\pi} e^{-i (r \tau - x \sin(\tau))} d \tau$$