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i´m preparing my exam on basic complex analysis, and i find out this exercise that i find nice, i need to show that

$$ \int_{|z|=1}exp\left(\frac{1}{z^k}\right)=\begin{cases} 2i\pi & k=1\\ 0 & otherwise\\ \end{cases} $$ i really don´t know how to deal this problem, i already know the tools but not how to use them, i know that i had Laurent´s series and residue calculus, the problem is that i don´t know how to calculate the residue, can you explain me how to proceed and how to find the residue. Thanks

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up vote 2 down vote accepted

Hint: Apply the residue theorem. Recall that the residue is the coefficient of the $z^{-1}$ term in the Laurent series. Take the Taylor series for $\exp(x)$ and substitute $x\mapsto z^{-k}$. That will give you the corresponding Laurent series.

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I'm not sure what you mean by that. There no need to "choose" the residue, the residue is the coefficient of $z^{-1}$ in the Laurent series. That's all it can be. – EuYu Oct 8 '12 at 1:01

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