Laurent series for exp[(a/2)(z - 1/z)] with |z| > 0

[Knightnole]

Hi. I am having trouble getting started on this problem.

I need to find the Laurent series for: f(z) = exp[(a/2)*(z - 1/z)] in |z|>0.

I know that the coefficients are: (1/2pi)*integral[cos(kx) - a*sin(x)]dx |(0 to 2pi)

But I am having trouble seeing how to get started on showing that this is true.

Thanks for any help anyone can offer.

 

[mutton]

Maybe this helps: $$z = e^{ix} \Leftrightarrow \frac{1}{2}\left(z - \frac{1}{z}\right) = i \sin x$$