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I was doing problems on complex integration and got stuck at one question. The question is

$$ \int_{\gamma}{{\rm e}^{{\rm i}\pi z}\left(z + i\right)^{2}\cos\left(nz\right) \over z^{2} - 1}\,{\rm d}z $$

where $\gamma= \left\{z: \left\vert\,z\,\right\vert=2\cos\left(\theta\right)\,, -\pi/2 \leq \theta \leq \pi/2\right\}$.

I am not getting any thought in which way to consider $\gamma$ in this question. Any hint will be sufficient for me. thanks a lot for help.

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    $\begingroup$ Remark: "doubt" (Indian English) = "question" (UK, US English). $\endgroup$ – GEdgar Dec 12 '13 at 1:26
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    $\begingroup$ @GEdgar: thanks, I would never have gotten that. $\endgroup$ – robjohn Dec 12 '13 at 1:30
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Hint: Notice that $$ 2\cos(\theta)(\cos(\theta),\sin(\theta))=(\cos(2\theta)+1,\sin(2\theta)) $$ is a circle of radius $1$ around $(1,0)$.

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  • $\begingroup$ @robjohn.... thanks for your timely help. will you please help in telling me how you got this idea about the circle. $\endgroup$ – monalisa Dec 12 '13 at 1:30
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    $\begingroup$ @monalisa: experience with certain polar curves reminded me of the polar equation for this circle. Then writing out $$r(\cos(\theta),\sin(\theta))=2\cos(\theta)(\cos(\theta),\sin(\theta))$$ I saw that the right hand side was $(\cos(2\theta)+1,\sin(2\theta))$ which is indeed a circle of radius $1$ around the point $(1,0)$. Did you also need help with the contour integral, or are you fine with that? $\endgroup$ – robjohn Dec 12 '13 at 1:34
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Hint: Think about the graph of $r = 2 \cos \theta$ in polar coordinates. The part of its graph with $-\pi/2 \leq \theta \leq \pi/2$ corresponds directly with the contour $\gamma$ (just replace the polar coordinate $(r, \theta)$ with the trigonometric number $z = r (\cos \theta + i \sin \theta)$.

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