Advanced Math/Improper integral
QUESTION: I am a meteorologist brushing up on some complex analysis and am stuck on this problem. The book gives the answer as Pi/200, but I can't see my error. I am trying to do this with residues. Thanks!
P.S. Excuse my horrible writing!
ANSWER: Thanks for the question. However, your writing is a little too unclear. Attached is an image with what I think is the integral you're working on.
I'm a little unsure why you need to use residues since the integral appears to be along the real axis while the poles are along the imaginary. Please clarify.
[an error occurred while processing this directive]---------- FOLLOW-UP ----------
QUESTION: That's it. The problem was to solve it by residues.
Got it. Using residues will work since the denominator has x raised to a power ≥ 2 than the numerator (so that the contour extending out to ∞ -> 0). Fair enough.
Your notes indicate that you correctly found the residue at the simple (i.e., order 1) pole at z1 = 3i (let me use z to represent the complex variable), namely by calculating the ratio of the numerator of the function (z^2) divided by the derivative of the denominator. Looks like you got z1 = i3/50.
The residue at z2 = 2i is more complicated to calculate since it is order 2. There are a lot of missing steps in your notes so I'm not sure is you are using the right method (that is to say, it isn't just a matter of checking algebra). Since it is not a simple pole, the technique you used for z1 won't work. Please let me know how you are approaching the calculation for this 2nd pole. Its a textbook exercise but the steps may be a little tricky. I'd be happy to look them over for you.