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Homework Statement
Find the Laurent series of f(z) = exp(1/z)/(z-1) around 0, and find Res{f(z), 0}.
Homework Equations
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] zn.
Then, using Cauchy's product, and rearranging some terms, f(z) = -[itex]\sum_{n=0}^∞[/itex] z-n [itex]\sum_{k=0}^n[/itex] z2k/(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?