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Math Help - Delta Functions and Integration

  1. #1
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    Delta Functions and Integration

    <br />
\frac{1}{2pi} \int k^2\frac{\sin \left(kr\right)}{kr} \frac{\sin \left(kr'\right)}{kr'} {dk}<br />

    (Couldn't figure out how to make the pi symbol, but that is a pi in the denominator at the beginning. Also the limit is from 0 to inf.). I'm supposed to show this integral equals 1/(4*pi*r*r') times the delta function of (r-r').

    I solved the integral and ended up with

    <br />
\frac{1}{4(pi)^2rr'} \left(\frac{sin\left(k\left(r-r'\right)\right)}{r-r'} - \frac{sin\left(k\left(r+r'\right)\right)}{r+r'}\ri  ght)<br />

    With k ranging from 0 to infinity. Now my problem is, I'm not sure exactly how to get a delta function out of this result. I can sort of see how it might emerge on the right sided term, but I'm confused as to how to express that mathematically. Also there is apparently a pi term that comes out as well when I compare it to the expected answer. Can anyone help me?
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by blorpinbloo View Post
    <br />
\frac{1}{2pi} \int k^2\frac{\sin \left(kr\right)}{kr} \frac{\sin \left(kr'\right)}{kr'} {dk}<br />

    (Couldn't figure out how to make the pi symbol, but that is a pi in the denominator at the beginning. Also the limit is from 0 to inf.). I'm supposed to show this integral equals 1/(4*pi*r*r') times the delta function of (r-r').

    I solved the integral and ended up with

    <br />
\frac{1}{4(pi)^2rr'} \left(\frac{sin\left(k\left(r-r'\right)\right)}{r-r'} - \frac{sin\left(k\left(r+r'\right)\right)}{r+r'}\ri  ght)<br />

    With k ranging from 0 to infinity. Now my problem is, I'm not sure exactly how to get a delta function out of this result. I can sort of see how it might emerge on the right sided term, but I'm confused as to how to express that mathematically. Also there is apparently a pi term that comes out as well when I compare it to the expected answer. Can anyone help me?
    Your integration is correct so far, except you should have  \pi and not  \pi^2 in your denominator.
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  3. #3
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    "Couldn't figure out how to make the pi symbol" Doubleclick on chiph588's pi to see how it's done.
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  4. #4
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    Quote Originally Posted by wonderboy1953 View Post
    "Couldn't figure out how to make the pi symbol" Doubleclick on chiph588's pi to see how it's done.
    And then say "Oh, of course!"
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  5. #5
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    There's probably a rule against bumping but I'm still curious about this problem and can't figure out how the delta function emerges from this integration. That should be a pi squared in the original integral so my integration is correct. Something about k ranging from 0 to infinite makes this integration look identical to the delta function. If anyone has any information they can contribute I'd appreciate it.
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