(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the Laurent series of f(z) = exp(1/z)/(z-1) around 0, and find Res{f(z), 0}.

2. Relevant equations

3. The attempt at a solution

To find the Laurent series, I wrote exp(1/z) = [itex]\sum_{n=0}^∞[/itex] z^{-n}/n!, and 1/(z-1) = -[itex]\sum_{n=0}^∞[/itex] z^{n}.

Then, using Cauchy's product, and rearranging some terms, f(z) = -[itex]\sum_{n=0}^∞[/itex] z^{-n}[itex]\sum_{k=0}^n[/itex] z^{2k}/(n-k)!

But I am unable to express this as a proper power series, with a closed expression for each coefficient, and thus find c_{-1}.

Since I'm unable to find c_{-1}directly, I found that Res(f, 1) = e, and Res(f, ∞) = -1, from which Res(f, 0) = 1-e. Is this correct?

How do I find the correct Laurent series?

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# Laurent series of exp(1/z)/(z-1)

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