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Integration of dirac delta composed of function of integration variable

  1. Dec 18, 2009 #1
    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
     
  2. jcsd
  3. Dec 18, 2009 #2
    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]
     
  4. Dec 18, 2009 #3

    diazona

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    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.
     
  5. Dec 19, 2009 #4
    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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