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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.

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    you had the cos multiplying the exponential, it has been fixed2012-11-08

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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...

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    Yes, the first $\exp$ in $\exp(r \exp(ix))$.2012-11-08
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HINT

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

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This integral can be obtained in closed form. I have written the complete answer on Quora. The link is posted below.

http://www.quora.com/How-do-I-solve-the-integral-int-limits-pi-_-0-e-cos-x-cos-sin-x-mathrm-d-x

I am posting the method below for $r=1$. The exact same steps can be take for any real $r$.

This integral certainly exists. Let's begin by rewriting $\cos(\sin x) = \frac{e^{i \sin x}+e^{-i\sin x}}{2},$ obtaining $\int_{-\pi}^{\pi} e^{\cos x}\cos(\sin x)~\mathrm{d}x = 2\int_0^{\pi} e^{\cos x}\cos(\sin x)~\mathrm{d}x = 2\,\Re \left[\int_0^{\pi} e^{e^{i x}}~\mathrm{d}x\right].$ Consider the integral $\int_0^{\pi} e^{e^{i x}}~\mathrm{d}x=\int_0^{\pi} \sum_{n=0}^{\infty} \frac{e^{inx}}{n!}~\mathrm{d}x = \sum_{n=0}^{\infty} \int_0^{\pi}\frac{e^{inx}}{n!}~\mathrm{d}x$ $=\int_0^{\pi}~\mathrm{d}x + \sum_{n=1}^{\infty} \int_0^{\pi} \frac{e^{inx}}{n!}~\mathrm{d}x$ $= \pi - i \sum_{n=1}^{\infty} \frac{(-1)^n-1}{n^2 (n-1)!}$ The summation can be obtained in closed form in terms of the hyperbolic sine integral or $\mathrm{Shi}(z)$ at $z=1$. However it is not required, as we are interested in the real part alone. Thus, $\int_{-\pi}^{\pi} e^{\cos x}\cos(\sin x)~\mathrm{d}x = 2\pi$

Cheers!

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    Can you please include the contents of the post you link to, or at least a sketch of it, in your answer? Link only answers are generally unwelcome here. If the link goes away, such an answer becomes entirely useless - although that is probably not likely to happen in this case. Still, we prefer to have the contents on site, and link-only answers run a risk of being deleted.2015-05-27
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you should first solve the integral of exp(rexp(ix)) by using Laplace transform of this exponential function then you take real part of this complex integral and the problem is solved :D