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# Fourier Type Integral using residue theorem watch

1. I'm trying to compute

f(x)=xsinx/(1+x^2) using the residue theorem.

I can get the answer which is pi/e however I'm struggling with one stage of my working and I want it to be correct.

I use the integral of ze^(iz)/1+z^2 to compute it over a semi circle in the top half of the complex plane. The semi circular part goes to zero due using Jordans Lemme, and now I'm stuck with the straight line part of the integral

Which is in the limit r goes to infinity and the integral of xe^(ix)/1+x^2 between -r and r.

The part I'm struggling to show is that we do not need to use the cos part of the exponential. I know it goes to zero but I don't understand why. Is it because of when you do a power expansion you get x(1+x...)/(1+x^2) which goes to one? Could you help pls.
2. (Original post by coheed94)
I'm trying to compute

f(x)=xsinx/(1+x^2) using the residue theorem.

I can get the answer which is pi/e however I'm struggling with one stage of my working and I want it to be correct.

I use the integral of ze^(iz)/1+z^2 to compute it over a semi circle in the top half of the complex plane. The semi circular part goes to zero due using Jordans Lemme, and now I'm stuck with the straight line part of the integral

Which is in the limit r goes to infinity and the integral of xe^(ix)/1+x^2 between -r and r.

The part I'm struggling to show is that we do not need to use the cos part of the exponential. I know it goes to zero but I don't understand why. Is it because of when you do a power expansion you get x(1+x...)/(1+x^2) which goes to one? Could you help pls.
I think this or very similar is here
http://www.thestudentroom.co.uk/show....php?t=3330283

or look in my site
3. Cheers buddy, there is a question that is similar, however you've just separated the real and imaginary parts and put them equal to the value you get. Is there any way to show that xcosx/1+x^2 goes to zero as r tends to infinity?
4. (Original post by coheed94)
Cheers buddy, there is a question that is similar, however you've just separated the real and imaginary parts and put them equal to the value you get. Is there any way to show that xcosx/1+x^2 goes to zero as r tends to infinity?
it is an odd function from -R to R !!

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