MHB Contour integral (please check my solution)

aruwin
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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.
 

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That looks correct. You could simplify the answer by noticing that $\sin\pi = 0.$
 
Opalg said:
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?
 
aruwin said:
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 $e^{i\pi/2} = i.$
 
Opalg said:
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?
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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