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Integral depending on coordinate differences

  1. Nov 21, 2012 #1
    1. The problem statement, all variables and given/known data
    Consider a function which depends only on a difference between two variables, and integrate it with respect to both:
    [tex]
    \int_a^b \int_a^b f(x-y)\, dxdy
    [/tex]
    Is there any way to simplify this expression, like reducing it into a 1-D integral?
     
  2. jcsd
  3. Nov 21, 2012 #2

    Dick

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    Use a change of variables like u=x-y, v=x+y. That will reduce it to a single integration over u after you do the dv integration.
     
  4. Nov 21, 2012 #3
    This gives me
    [tex] dxdy = (dv^2-du^2)/4 [/tex]
    and I don't see how this makes the integral any easier
     
  5. Nov 21, 2012 #4

    Dick

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    That's not how you do change of variables in double integration. dxdy is equal to dudv times a Jacobian factor, remember? http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant Then you have to change the limits.
     
  6. Nov 21, 2012 #5
    Oh, thanks Dick, I wasn't aware of this...
     
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