# Fourier series of $e^{\cos x}$

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 but i cannot go forward more than expanding $e^{\cos(t)}$. Could any of you help me with a hint?

• I think this is a question for math.stackexchange – Michael Freimann Jun 18 '17 at 20:59
• That's not entirely fair; the coefficients are not elementary. As I recall the Fourier coefficients of $\exp(\cos t)$, and more generally $\exp(c \cos t)$, can be expressed in terms of Bessel functions. – Noam D. Elkies Jun 18 '17 at 21:33

$$\int_0^{2\pi} \exp(int) \exp(\cos(t))\; dt = \int_{-\pi}^{\pi} \cos(n t) \exp(\cos(t))\; dt = 2 \pi I_n(1)$$ where $I_n$ is a modified Bessel function of the first kind and thus $$I_n(1)=\frac12\sum_{k\geq0}\frac1{4^kk!(n+k)!}.$$