Integration of dirac delta composed of function of integration variable

  • Thread starter kmdouglass
  • Start date
Hi all,
I'm working through Chandrasekhar's http://prola.aps.org/abstract/RMP/v15/i1/p1_1" [Broken] 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
 
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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

Homework Helper
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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.
 
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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