Integration of dirac delta composed of function of integration variable

[kmdouglass]

Hi all,
I'm working through Chandrasekhar's http://prola.aps.org/abstract/RMP/v15/i1/p1_1" and can not understand the steps to progress through Eq. (66) in Chapter 1. The integral is:

\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}}

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

 

[phsopher]

Weird, I didn't get that one either. I got

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

 

[diazona]

Weird, I didn't get that one either. I got

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

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]

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