Finding the Laurent Series for f(z): A Problem in Need of Help

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SUMMARY

The discussion focuses on finding the Laurent series for the function f(z) = exp[(a/2)*(z - 1/z)] in the region |z| > 0. The coefficients for the series are defined as (1/2π) * integral[cos(kx) - a*sin(x)]dx from 0 to 2π. The user expresses difficulty in understanding how to derive the series from these coefficients. A transformation using z = e^{ix} is suggested to aid in the analysis.

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

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