(adsbygoogle = window.adsbygoogle || []).push({}); 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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# Complex differential 1-form question

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