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 .
Let . By using a semi-circle contour with we find:
.Now take limit , the middle integral is zero (by Jordan's lemma) while the RHS integral is computed by residue theorem:
.It remains to compute note that this is a simple pole:
.Thus,
.This mean by equating real and imaginary parts,
.