# 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" [Broken] 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
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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#### 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

Homework Helper
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

"Integration of dirac delta composed of function of integration variable"

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