Complex analysis integrals (1 Viewer)

[integral95]

Use complex analysis methods to find

 

[integral95]

.

 

[glittergal96]

Use complex analysis methods to find

Which parts are confusing you?

For the first one you want to change that real integral into a contour integral with circular contour, by taking z=e^(i*theta), and this reduces the problem to integrating a rational function over the counterclockwise unit circle.

The denominator can be factorised easily and we have poles inside the contour at z=0,1/2 of orders 3 and 1 respectively.

Computing residues at these poles is the next step, I did the first by a series expansion and the second is easy because it is a simple pole.

The residue theorem then lets us evaluate the integral. You should get something like 27*pi/8-21*pi/8=3pi/4 as your answer.


For the second, I don't know if you typoed the question but you probably want to take your branch cut on the positive REAL axis. Then, we take a keyhole type contour and show that the integral on the circular arcs tend to zero (triangle inequality should accomplish this for the outer one, and the inner one is pretty trivial). We finally express the original integral I in terms of the residues of f at the simple poles at +-2i, and compute these residues.

The answer is pi*2^(-1/4)*sin(pi/8) if I calculated things correctly.


Do the above methods make sense to you? If so, try and do the calculations yourself for practice, but I can help you further if you have any troubles with this. (It's just a bit tedious for me to write things out fully right now.)