Use the Laurent series to prove that $\int_{0}^{2 \pi} e^{\cos(t)} \cos(nt-\sin t)\,dt=\frac{2 \pi}{n!}$

Question

Need help to demonstrate the next integral

$\int_{0}^{2 \pi} e^{\cos (t)} \cos(nt-\sin t)\,dt=\frac{2 \pi}{n!}$, $n=0, \pm 1,...$

I have tried using Laurent’s development of $e^{1/z}$, but i have not managed to prove it.

Now, i have to

$$\frac{1}{n!} = \frac{1}{2i\pi} \int_{C_r} e^{\frac{1}{z}}z^{n-1} dz.$$

I have tried to use the parameterization of a circle, but i have terms that do not correspond to my problem

Answer 1

The trick is your choice of contour, $|z|=1$ with $z:=e^{it}$, obtains $\oint z^{j-1}dz=2\pi i\delta_{j0}$, so$$\oint z^{n-1}e^{1/z}dz=\oint\sum_{k\ge0}\frac{1}{k!}z^{n-k-1}dz=\frac{2\pi i}{n!}.$$