How do I show that $\exp(\frac{c}{2}(z-\frac{1}{z}) = \sum(a_n\cdot z^n)$ where $a_n = \frac{1}{2\pi} \int \cos(n\theta - c \sin(\theta)) d\theta$

Question

This is what I have so far I have written the question on the picture. (Sorry I did not know how to add integral and summation signs properly on the question)

Answer 1

On the unit circle $z=e^{i\theta}$. Hence,

$$\begin{align} \frac1{2\pi i}\oint_{|z|=1}\frac{e^{c(z-z^{-1})/2}}{z^{n+1}}\,dz&=\frac1{2\pi}\int_0^{2\pi} e^{-in\theta}e^{ic \sin(\theta)}\,d\theta \end{align}$$

Now exploit the $2\pi$-periodicity and even-odd symmetries. Can you finish now?