MHB The Residue Theorem To Evaluate Integrals

[joypav]

View attachment 7330

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!

 

[Euge]

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.

 

[joypav]

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.

 

[Euge]

It looks right so far. Now show that the integral of $f$ becomes negligible over certain parts of the contour.

 

[joypav]

Thanks!

 

[Euge]

For completion, the integral evaluates to

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