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I'm trying to study for my final this Thursday and for some reason the problem that is giving me the most trouble is, according to the professor, the "easiest one". I'm just not seeing something here, it's been a long time since I've done anything in complex that looks like this. The problem is:

$$A = \left\{ \zeta: |\zeta|= \frac{1}{2}\right\}, B: = \left\{\omega : |\omega|< \frac{3}{4}\right\}. \text{ Define } f(w) : = \oint_A \frac{z(z+1)}{z^2 + 2z -w} dz.$$

I'm asked to show that $f$ is analytic in $B$ and to find $f'(0)$. I need to learn the idea behind this much more than I need an answer to this specific question, so hints or related questions are much appreciated.

Thank you fixed!

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    It seems like there may be something missing in the problem. First, you haven't said what the contour you're integrating over is. Second, I don't see where the set $A$ comes into the problem. The question never makes any mention of $A$.2012-12-12
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    Are there any poles in the interior of $A$?2012-12-12
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    This seems like a hard question to answer, as it depends on what $\omega$ we are evaluating at. The for example, if $\omega = \frac{2}{3}$ then there is a pole at $z = \frac{1}{2} ( \sqrt{7} - 2)$. But of course there will be different poles for different $\omega$. Are you suggesting I do something with the residues as a function of $\omega$?2012-12-12

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