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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!

improper integral
improper integral  
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.

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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.

Good luck!

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randy patton


college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography


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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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