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I'd really like some help with this problem. I'm supposed to find $$ \int_{\partial B_{2}(0)} \frac{1}{(z^n-1)^2}dz,$$ where $B_2(0) = \{ z \in \mathbb{C} \; | \; |z|<2 \}$ (ie. the ball of radius 2 and centered at 0). This, of course, amounts to finding the residues. I can do it by investigating the derivative $(\text{Res}(f, z_i) = \frac{d}{dz}\left( \prod_{j \neq i} \frac{1}{(z-z_j)^2} \right)$, where I'm denoting $z_i$ the $i$-th root of unity), but it looks kind of messy attacking it straight on. Any ideas?

Also, in a later item in the same question I should be able to find a primitive for $\frac{z^4}{(1-z^3)^2}$, where $|z|<1/2$. I have absolutely no idea here (at an intermediary item, we have to find $\int_{\partial B_{1/2}(0)} \frac{z^{2n-2}}{(1-z^n)^2}$, but this is zero since there are no residues, and this calculation doesn't seem to help...). Can you give me some direction?

Thanks!

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    Yes, sorry, I meant the boundary of the ball. Thanks for correcting my mistake, Jorge.2012-12-09
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    Hint: Making the substitution $w=1/z$ will improve your life considerably, except possibly when $n=1$.2012-12-09
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    Gerry, guess I made it clear now: $B_2(0)$ is a common notation for the ball of radius 2 centered at 0.2012-12-09
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    Micah, that really helps in connecting the integrals $\int_{\partial B_2(0)} \frac{1}{(z^n-1)^2}$ and $\int_{\partial B_{1/2}(0)}$, but I can't see why this calculation would be useful to find the value of either or to find a primitive of $\frac{z^4}{(1-z^3)^2}$...2012-12-09
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    The integral you get after substituting ought to involve a holomorphic function which is considerably better behaved on the interior of its contour (it will be holomorphic except possibly at zero, because your original function was holomorphic on the exterior of its contour except possibly at infinity). That wasn't intended to be a hint for the second half, but if your function is poleless in $B_{1/2}(0)$ its integral is path-independent...2012-12-09

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