- #1
Finnjessy
- 1
- 0
Homework Statement
Hello all. I'm currently attempting to prove the central limit theorem using a simple case of two uniformly distributed random variables. Aside from being able to solve it using convolutions, I also wish to solve it by using the Dirac Delta function. That aside, the integral I'm trying to solve is
[tex] \int_{-\infty}^{\infty} \! -\frac{e^{ikx}}{k^2} \, \mathrm{d}k [/tex]
with [itex] x \in [0,2] [/itex]
Homework Equations
The Cauchy Residue Theorem
[tex] Res(f,c) = \frac{1}{2\pi i}\oint_\gamma \! f(z) \, \mathrm{d}z = [/tex]
The Attempt at a Solution
According to Plemelj's theorem if we have a simple pole on the contour of our chosen integral, the result is equal to [itex] \pm i\pi Res(f,c) [/itex], where c is the location of the pole.
I know from both a Mathematica calculation and a Monte Carlo simulation that the solution to this integral is in fact [itex]x\pi[/itex]. However, if I calculate the residue of the function and use Plemelj's theorem, I get the exact result that I would want. The problem is that as far as I know, Plemelj's theorem should only work for simple poles and the second order term should change this calculation. My question therefor is, why does the theorem work for my second order pole?
As far as alternate ways of calculating the integral, I've tried several methods but haven't found anything that would work for second order poles. I've also attempted several different substitutions to get rid of the second order pole, but all seem to blow up to infinity or go to zero. So assuming that Plemelj's theory indeed can't be applied to higher order poles, I'm kindly asking for help as far as to what direction I should head with the calculation. Thank you.
Last edited: