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Change of variables (i don,t understand)

  1. Jul 16, 2005 #1
    let be the integral:

    [tex]\int_1^{\infty}\int_{-\infty}^{\infty}f(x,y)dydx [/tex]

    i make the change of variable xy=u y=v whose Jacobian is 1/v but then what would be the new limits?...
     
  2. jcsd
  3. Jul 16, 2005 #2

    OlderDan

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    v is y, so u = xv, and x can never be less than 1. What does that tell you about the possible values of u for any given v?
     
  4. Jul 17, 2005 #3
    could you write the new limits...i can,t work it out the new values of v for v gives me (-8,8) (here 8 means infinite but for u i got... (0,8) is that true?..what would happen if y choose the change of variable x/y=u y=v? thanx
     
  5. Jul 18, 2005 #4

    OlderDan

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    From your original integral

    [tex]\int_1^{\infty}\int_{-\infty}^{\infty}f(x,y)dydx [/tex]

    you have y ranging over all reals and x ranging from +1 to infinity. With v = y, v will range over all reals and with u = xy = xv, u will range from -infinity to v when v is negative and from v to infinity when v is positive. Looks like that gives you

    [tex]\int_{-\infty}^{0}\int_{-\infty}^{v}g(u,v)dudv + \int_{0}^{\infty}\int_v^{\infty}g(u,v)dudv [/tex]

    g(u,v) includes the contribution from f(x,y) and the Jacobian, and you need to be careful with the signs.
     
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