Residue Theorem

[rubba61]

I am having trouble evaluting the integral from -inf to inf of x*sin(x)/(1+x^4) using residues and closed contour integration in the complex plane. I know how to find the residues of 1/1+z^4 and that I only need to use two of them to represent the two poles in the upper half of the complex plane. :duh: However I get a bit confused with the sin and x terms in the numerator. Any ideas about those residues and how they would interact with the residues of the denominator or if the residues of the denominator even have to be considered would be helpful.

[matrix1329]

Does x*sinx have any poles in the complex plane? I don\'t think so: Take z*sinz. z obviously has no poles or branchcuts and sin z has neither:

http://mathworld.wolfram.com/Sine.html

So the only poles you have to deal with are the ones from the denominator.

René

Originally posted by rubba61
I am having trouble evaluting the integral from -inf to inf of x*sin(x)/(1+x^4) using residues and closed contour integration in the complex plane. I know how to find the residues of 1/1+z^4 and that I only need to use two of them to represent the two poles in the upper half of the complex plane. :duh: However I get a bit confused with the sin and x terms in the numerator. Any ideas about those residues and how they would interact with the residues of the denominator or if the residues of the denominator even have to be considered would be helpful.

[Clem]

Don\'t keep sine. Make it Im\\integral e^{iz}.

[Parseval]

Does any one have a good reference for these types of calculations
My graduate course in Mathematical Physics used Boas, which was pretty good, but I got a little lost. Is there any better books for residue and contour integrals and all that complex theory for a physicist?
I\'m gonna need to know how to do this stuff eventually.

[nmondal]

We did use Churchill & Brown.
http://www.campusi.com/bookFind/asp/bookFindPriceLst.asp?prodId=0072872527
Now I use some Russsin book, that is utterly brilliant...
:cool:

[Clem]

Arfken or Butkov are higher level than Boas, and are written for physicists.

[SamuelPrime]

Clem\'s suggestion to replace sin x by exp(ix) helps since your integral is
just the real part of the integral of the same function you have but with
sin x replaced by exp(ix). That\'s step 1.

The next step is look at the simple closed coutour in the complex plane
given by the interval [-R,R] followed by the semi-circle of radius R
(parametrized by R exp(it) where t goes from 0 to pi). Now this contour
surrounds two singularites of your function (each of which is a simple
pole). Their residues are easy to find. So now you have been able to
find the integral over this closed contour. When you break it down you
get two integrals, one over [-R,R] and the other is over the semicircle.
The latter one you can easily check goes to 0 as R goes to infinity. That
should give you the integral you want. I hope that helps.

[Edited on 6-13-2005 by SamuelPrime]

[lonelyphysicist]

x*sin(x)/(1+x^4)
= Im[x*exp(i x) / ( (1+ix^2)(1-ix^2) )]
= Im[x*exp(i x) / ( (1+exp(i pi/4)x)(1-exp(i pi/4)x) (1+exp(-i pi/4)x)(1-exp(-i pi/4)x) )]

To do the integral we close the contour on the upper half (+ve imaginary) plane because the exponential damps out and kills the integral along the infinite semi-circle. That means the contour integral will yield the same value of the integral along the real axis.

The poles in the positive imaginary half of the plane are simple because x*exp is entire; they are: exp(i pi/4) and -exp(-i pi/4). The rest of the steps simply involve summing the residues x (2 pi i) and taking the imaginary part, with the residue given by the evaluation of the integrand at the pole w/o the term in the denominator that makes things go kaboom.

[paytexti]

http://www.math.sunysb.edu/~vkiritch/p3.pdf
ı need to your help obout the firt integral at the this link..
pls help me!!!!!!

[nox]

It should be pi/(2*sqrt(3)). ... but it sounds like a homework problem. I suggest you to show us what you've tried to solve it, next time.

[paytexti]

that is my exam question couldnot sove it at exam and ı want to see the solution of that problem

[Clem]

1. To get the poles of the function (the zeros of the denominator),
solve z^4+z^2+1=0 for z^2 (two complex roots).
2. Write each root in polar form to get z=(z^2)^{1/2}.
3. There will be two poles in the UHP. Just do the contour integral then.