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For my homework assignment I've been given a number of complex integrals to solve. I've already asked for help on a specific example here, but I was somewhat dissatisfied with the answers. The answers on that question simply answered the example I posted, but did not provide any insight into the problem solving strategy. The example I am looking at now is:

$$ \int_\frac{\pi}{2}^{\frac{\pi}{2} + i} \cos \; (2z) \; \mathrm{d}z $$

According to the advice on my last post, I should parametrize this. So, let

$$ \gamma(t) = \frac{\pi}{2} + it \text{ for } 0 \leq t\leq 1 $$

This is where I am lost. How can I convert my integral into a form which I know how to integrate? Any help would be great. Thanks.

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1 Answer 1

The function being integrated is analytic in a region containing the endpoints. Therefore, it doesn't matter what path you choose within such a region. To wit, just integrate:

$$\int_{\pi/2}^{\pi/2+i} dz \: \cos{2 z} = \frac{1}{2} \left[\sin{(\pi + 2 i)} -\sin{\pi}\right] = -\frac{i}{2} \sinh{2}$$

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