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-   -   Integration of dirac delta composed of function of integration variable (http://www.physicsforums.com/showthread.php?t=364332)

kmdouglass Dec18-09 12:59 PM

Integration of dirac delta composed of function of integration variable
 
Hi all,
I'm working through Chandrasekhar's Stochastic Problems in Physics and Astronomy and can not understand the steps to progress through Eq. (66) in Chapter 1. The integral is:

[tex]\prod^{N}_{j=1} \frac{1}{l^{3}_{j}|\rho|}\int^{\infty}_{0} sin(|\rho|r_{j})r_{j}\delta (r^{2}_{j}-l^{2}_{j})dr_{j} = \prod^{N}_{j=1} \frac{sin(|\rho|l_{j})}{|\rho|l_{j}}[/tex]

Could anyone show the steps on how this result was obtained? I am aware of how to simplify a dirac delta that is composed of a function, but it does not lead me to the above result. Thanks.

-kmd
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

phsopher Dec18-09 06:51 PM

Re: Integration of dirac delta composed of function of integration variable
 
Weird, I didn't get that one either. I got

[tex]\prod^{N}_{j=1} \frac{sin(|\rho|l_{j})}{2|\rho|l_{j}^3}[/tex]

diazona Dec18-09 10:07 PM

Re: Integration of dirac delta composed of function of integration variable
 
Quote:

Quote by phsopher (Post 2498164)
Weird, I didn't get that one either. I got

[tex]\prod^{N}_{j=1} \frac{sin(|\rho|l_{j})}{2|\rho|l_{j}^3}[/tex]

That seems more reasonable. In the equation posted by the OP, the units are inconsistent between the two sides, so it can't be right.

kmdouglass Dec19-09 01:21 AM

Re: Integration of dirac delta composed of function of integration variable
 
Yes, you are right about the units. And someone else aside from myself got phsopher's result as well.

A few equations back, the author defines the probability distribution that he is using, and if I integrate over all angles and radial distances, I don't get unity. I think there are significant typos in this section. Thanks for the help.

kmd


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