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Integrating a dirac-delta

  1. Jan 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Show:

    02∏ δ(sin θ - k) dθ

    equals

    θ(1-|k|) * 2/sqrt(1-k2)

    2. Relevant equations

    integrating dirac-delta function

    3. The attempt at a solution

    [sin (2pi) - k] - [sin 0 - k] ??
     
    Last edited: Jan 30, 2012
  2. jcsd
  3. Jan 30, 2012 #2

    lanedance

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    now I'm assuming the second theta is actually a heaviside function (some bad choice of notation but surprisingly to a few other recent posts)

    as for your answer it doesn't make a heap of sense, what do you actually know about integrating the delta function...
     
    Last edited: Jan 30, 2012
  4. Jan 30, 2012 #3

    lanedance

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    first consider when |k| > 1,what is the value of the integral...?
     
  5. Jan 30, 2012 #4

    lanedance

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    now when |k| < 1, i would try a substitution... and ask yourself again what do you actually know about integrating the delta function?

    the form of the answer may also give u a hint, though I haven't fully worked it through... what common function f(k) integrates to 1/(1-k^2)?
     
  6. Jan 31, 2012 #5
    I am not sure how to integrate dirac-delta of a polynomial such as [sin(theta) - k] unfortunately.
     
  7. Jan 31, 2012 #6

    HallsofIvy

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    Do you know what [itex]\int_a^b \delta(x) dx[/itex] itself is?
     
  8. Jan 31, 2012 #7

    lanedance

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    exactly start from what you do know
     
  9. Jan 31, 2012 #8

    lanedance

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    also that is not a polynomial
     
  10. Jan 31, 2012 #9

    lanedance

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    Once you can answer that ask the following questions, how do you solve
    [itex]\int_a^b \delta(2x) dx[/itex]

    then
    [itex]\int_a^b \delta(2x-k) dx[/itex]

    and what is the value of
    [itex]\int_a^b \delta(2x-k) f(x) dx[/itex]

    If you can do those you shoudl have a pretty good idea how to do approach the problem, along with some trig and you should be there
     
  11. Jan 31, 2012 #10
    okay, i think i got it, or at least as long as I am assuming a trig identity correctly, much thanks
     
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