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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)

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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?

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  • $\begingroup$ @stanleycho Happy Holidays! Please let me know how I can improve my answer. I really want to give you the best answer I can. $\endgroup$
    – Mark Viola
    Dec 23, 2021 at 4:52

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