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Gaussian Integral Help

by TheMightyJ
Tags: gaussian, integral
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Jun26-09, 05:39 AM
P: 8
1. The problem statement, all variables and given/known data

Basically, i have to find the solution to:

Int( x2 * exp (-(x-w)^2) , x= -infinity.. infinity)

2. Relevant equations

I realise this is connected to Gaussian Integration. So that if i have Int (exp(-x2), x=-infinity ... infinity) the answer is sqrt(Pi)

Also, i have read that there is a trick to solving such an integral.

you would consider F(a) = Int exp( - a * x^2) ( = sqrt(pi/a) ).

and then int dx x^2 exp(-x^2) = - F'(a) for a=1.

i understand this.

3. The attempt at a solution

Basically, the above is how i have attempted to work a way towards the solution, the trouble i am having is with a substitution i realise i must make at some point. at some point i must have x = x - w

But how do i implement this substitution??

This is my first post here, so hopefully that was somewhat clear and the relevant info is there. Thanks for any help.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
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Jun26-09, 06:06 AM
P: 550
I don't know if this is the easiest solution, but you can try u = x - w. The integral splits into three integrals, which can be solved easily.

u = x-w
du = dx

Then, x = u+w, so x^2 = (u+w)^2, and the integral becomes:
[tex] \int (u+w)^2 e^{-u^2}\,du[/tex]
Jun26-09, 06:09 AM
P: 8
Will give that a try! thank you very much!

Jun26-09, 06:30 AM
P: 8
Gaussian Integral Help

Yep, tried it and it worked, brilliant, thank you!
Jun26-09, 07:30 AM
P: 8
Okay, so there is another question, similar in some ways to the previous one, but i am also having trouble. Mainly im stuck on where to start!

The integral this time is

Int ( e-(x-w)2 - j*x dx

the limits of integration are again, -infinity to infinity.

obviously there is some sort of gaussian integral stuff going on, but that "-j*x" has thrown me off, how should i deal with this?

just an idea to help me get started would be muchly appreciated, thanks!
Jun26-09, 07:32 AM
HW Helper
P: 1,495
You need to complete the square in this case. That is write (x-w)^2+jx as (....)^2+constant.
Jun26-09, 07:33 AM
P: 8
Obviously. Thank you! really not spotting obvious techniques today. Thanks for the help!
Jun26-09, 07:36 AM
P: 550
j is not the imaginary number j (or i) right? If it is, that makes matters a little different.
Jun26-09, 07:37 AM
P: 8
no no, j is just the coefficient of x. Thanks.

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