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# Advanced Math/Integral with residue theory

Question
QUESTION: Hi Randy, I am a meteorologist with a degree in math as well as a degree in meteorology degree. I have an integral that I thought I knew how to do, but according to my calculator, I made an error somewhere, but can't find my error. The integral is the integral of sin(2x)/(x^2+9) dx from 0 to infinity.

I start by defining the function phi(x)=e^(2i*x)/(x+3i)^2.
I take the derivative of phi and get (e^(2i*x)/(x+3i)^2)(2i-2/(x+3i))

Now, I evaluate the derivative at 3i and get -7ie^-6/108 so the integral is pi*i*-7i*e^-6/108=7pi*e^-6/108 or approximately .00051. My calculator gives me .00709. Can you see where I went wrong?

Thanks!

ANSWER: David, I'm afraid I don't follow your approach. For doing a contour integral, I would have defined

sin(2x)/(x^2+9) = Im{phi(z)}

phi(z) = exp(i2x)/(z^2+9)

where z is complex,

and then defined a contour around the simple pole at z = 3i to include the x axis from 0 to infinity.

Please let me know what approach you are using as it may well be the right way. I'm also curious what your calculator does.

---------- FOLLOW-UP ----------

QUESTION: I don't have my book in front of me right now, but basically, it said to define phi such that it's analytic at 3i and then define the contour around z=3i. Also, the denominator should be squared; I left that part off on accident.

ANSWER: Thanks for the response. When you say the denominator should be squared do you mean the original function should be

sin(2x)/[x^2+9]^2  ?

I can try and work with that. I'd still like to see what your textbook, and in particular, your calculator says. BTW, in my calculations (with the un-squared denominator), I was getting the value to be zero. I don't see how squaring the denominator will change that, but we'll see.

---------- FOLLOW-UP ----------

QUESTION: Yes. That's right

contour integration ca

I know the denominator was supposed to be squared, but I worked out that one as well and came up with with basically the same result). I can show the the integral of sin(2x)/(x^2+9) or sin(2x)/[(x2+9)^2] over the whole real line is zero but it is hard to evaluate the integral from zero to infinity. I've tried various contours but have yet to figure it out.

Anyway, any insight as to what your textbook and/or "calculator" says would be appreciated. A quick integration using Excel shows the integral going to zero as R --> infinity, but this is not a proof.
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#### randy patton

##### Expertise

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

##### Experience

26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

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

Education/Credentials
M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

Past/Present Clients
Also an Expert in Oceanography