MHB The Residue Theorem To Evaluate Integrals

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The discussion focuses on using the residue theorem to evaluate integrals, highlighting the need for summation of residues and the importance of selecting the correct branch for multi-valued functions like square roots. Participants emphasize the necessity of choosing an appropriate contour, such as a keyhole contour, to apply the theorem effectively. The calculation involves computing four integrals and equating them to 2(pi)i times the sum of the residues. Additionally, it's noted that certain parts of the contour integral can be shown to be negligible. The final evaluation of the integral is given as π/(2√2)√(√5 - 1).
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I have no idea if this is in the right direction. I know I am going to need the summation of the residues to use the theorem. I found the residues using the limit, but do I need to change these using the euler formula?

We are supposed to be working problems at home and I am getting a bit lost as the semester goes on. I would really appreciate some help or a push in the right direction!
 

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Square roots are multi-valued, so you have to choose a branch $\log z$ first. Also, you need to choose a contour for which the residue theorem can be applied. Consider using a keyhole contour.
 
Okay, yeah. Something more like this? Lacking details of course, but is this the idea?
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Then I will have four integrals to compute, set equal to 2(pi)i*sum of the residues.
 

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It looks right so far. Now show that the integral of $f$ becomes negligible over certain parts of the contour.
 
Thanks!
 
For completion, the integral evaluates to

$$\boxed{\frac{\pi}{2\sqrt{2}}\sqrt{\sqrt{5}-1}}$$
 

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