# doubt in complex integration

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.

• Remark: "doubt" (Indian English) = "question" (UK, US English). – GEdgar Dec 12 '13 at 1:26
• @GEdgar: thanks, I would never have gotten that. – robjohn Dec 12 '13 at 1:30

Hint: Notice that $$2\cos(\theta)(\cos(\theta),\sin(\theta))=(\cos(2\theta)+1,\sin(2\theta))$$ is a circle of radius $1$ around $(1,0)$.
• @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? – robjohn Dec 12 '13 at 1:34
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 $z = r (\cos \theta + i \sin \theta)$.