hi all, my first post; had a minor headache with this problem lol.

PROBLEM 1:

Finding Residue:

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find Res(g,0) for

My Attempt/Solution:

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I know

so now

we know the residue is the coefficient of the -1th term (or the coefficient of ) but there is no -1th term as you can see. So does that mean the residue is 0, or am I missing something?

PROBLEM 2:

Finding Integral:

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Evaluate by integrating around a suitable closed contour:

My Attempt/Solution:

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First consider the intergral,

where and and

Now, let . The integrand has 2 simple poles, one at 2i and one at -2i. Only 2i is inside the contour so:

By Cauchy's Residue Theorem, we have:

To show that , we apply Jordan's Lemma : .

So now this means,

Please could you verify whether I have worked these out correctly. Thanks a great deal, highly appreciated .