Contour integral (please check my solution)

[aruwin]

Hello. Can someone check my solution for this question? I am not sure what to do about the "from e^-pi*1/2 to e^pi*i/2" part. I ignored that part.

 

[Opalg]

That looks correct. You could simplify the answer by noticing that $\sin\pi = 0.$

 

[aruwin]

That looks correct. You could simplify the answer by noticing that $\sin\pi = 0.$

Thanks. But what does "from e^-pi*1/2 to e^pi*i/2" mean?

 

[Opalg]

Thanks. But what does "from e^-pi*1/2 to e^pi*i/2" mean?

It means that $C$ is the contour starting at the point $e^{-i\pi/2} = -i$ on the unit circle, and ending at the point $e^{i\pi/2} = i.$

 

[aruwin]

It means that $C$ is the contour starting at the point $e^{-i\pi/2} = -i$ on the unit circle, and ending at the point $e^{i\pi/2} = i.$

Does that affect the calculation here? Is that why we integrate from -pi/2 to pi/2?