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Complex differential 1-form question

  1. Jul 17, 2011 #1

    kvt

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    1. The problem statement, all variables and given/known data
    I am trying to solve Nakahara Ex. 1.5. I have already solved part (1), but I am stuck trying to generalize the equation of (1) to prove part (2). I think I will be able to complete the proof if I can establish the following equation:


    2. Relevant equations

    [tex] \int dz d\overline{z} \exp({-z\overline{z}}) = \int dx dy \exp({- x^2 - y^2}) [/tex]



    3. The attempt at a solution
    Using [itex] z = x + iy [/itex], it is obvious that both exponents are the same, but the Jacobian from the coordinate transformation does not seem to be equal to 1. Is it true that [itex] dz d\overline{z} = dx dy [/itex] ? If so, why?
     
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  3. Jul 17, 2011 #2

    hunt_mat

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    Homework Helper

    What you have is [itex]dz\wedge d\bar{z}[/itex], compute [itex]dz[/itex] and [itex]d\bar{z}[/itex] and take their wedge product.
     
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