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March 8th, 2017, 02:50 PM   #1
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Contour Integration Problem

I'm struggling to evaluate this integral:

$\displaystyle \int_{0}^{2\pi} {\frac{d\theta}{7-2\cos \theta}}$

I have used the substitution $\displaystyle z=e^{i\theta}$ and obtained $\displaystyle d\theta = -i\frac{dz}{z}$ and substituted it in. I'm then left with the integrand equal to $\displaystyle \frac{1}{z^2 -7z +1}$ after taking the constant $\displaystyle i$ out the front of the integral. This left me with a pole inside the unit circle equal to $\displaystyle \frac{7-3\sqrt{5}}{2}$ and I'm not quite sure how to proceed after this. I tried using the residue formula for a simple pole but it's not giving me the correct answer which is given in the question and is $\displaystyle \frac{2\sqrt{5}\pi}{15}$.

So I think my method is wrong. Any help would be appreciated.
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March 8th, 2017, 03:40 PM   #2
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First, what is your contour integral and what is your contour?
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March 8th, 2017, 03:49 PM   #3
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I'm not given any more information. I thought I could use the contour as the unit circle centred at 0 since f(z) equal to the integrand given would be analytic inside the unit circle except at the pole calculated. Just generally a bit confused with the method.
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March 8th, 2017, 04:45 PM   #4
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did you multiply the residue by $2\pi i$ ?
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March 8th, 2017, 05:35 PM   #5
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I think I am calculating the residue wrongly. I'm using Res(f,c) for the calculated simple pole c is equal to the limit as z tends to c of (z-c)f(z) however I am getting 0 as an answer.
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March 8th, 2017, 06:09 PM   #6
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March 9th, 2017, 12:27 AM   #7
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Thanks a lot!
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