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Let $\Gamma$ be a circumference with center $3i$ and radius $5$ oriented counter-clockwise. Calculate: $\displaystyle\oint_{\Gamma} \frac{z}{(z^2 -2z)(z^2 - 4z + 13)} dz$

At a first glance I thought it was a line integral but then I realized the function $f(z)$ has only one unknown ($z$), while it should have 2 such that I can parameterized it. Any hint is appreciated.

Thanks,
rubik

P.S. Sorry for the english: the problem is a translation from another language and I don't know whether I used the correct terms.

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You need to use Cauchy Residue Theorem. In your case, the integrand has poles at $z = 0,2,2 \pm 3i$. The poles $0,2,2+3i$ (except $2-3i$) are all inside the circle over which you are integrating. Hence you need to compute the residue at all these poles and add them up and multiply by $2 \pi i$ to get the value of the integral.

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    @yzhao: Thank you for explaining an alternative approach: very interesting!2012-06-26