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Integral of x*delta((x/y)-t) dx from 0 to infinity

  1. Feb 20, 2012 #1
    1. The problem statement, all variables and given/known data

    ∫x*delta((x/y)-t) dx from 0 to infinity

    2. Relevant equations
    ∫x*delta((x/y)-t) dx from 0 to infinity = ty*|y|*θ(ty)

    3. The attempt at a solution

    Okay, so using the transformation of variables technique via the Jacobian, I see where the |y| comes from. However, using the dirac delta method I have NO clue how that |y| is derived logically. I know that the property of the dirac delta is that, for ex, ∫f(x)*delta(x-a) dx from x = -infinity to infinity = f(a). In other words, we solve the equation x-a = 0 for x. Likewise, we have ((x/y)-t) = 0, solved for x = ty.

    So I can see where ty*θ(ty) comes from...but how is the |y| derived?


    Thank you very much.
     
  2. jcsd
  3. Feb 20, 2012 #2

    vela

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    Try using the substitution u=x/y to get the argument of the delta function to look like it appears in the property you cited.
     
  4. Feb 20, 2012 #3

    Dick

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    Don't forget that if y is negative you need to adjust the limits in the u integral. That's where the absolute value will come from.
     
  5. Feb 20, 2012 #4
    Ok assuming y is positive( it actually is for the question I was really working on)...
    okay so u = x/y implies x = y*u. So we have (y*u*Delta(u-t))
    we need to convert dx to du
    x = y* u
    Dx = y du
    int(y*u*Delta(u-t)*ydu)
    So u = t and we get
    y*t*y*Heaviside(yt)




    I can see how the y is derived now. Hopefully my logic is perfect in my derivation. Thank you!!
     
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