# Complex differential 1-form question

1. Jul 17, 2011

### kvt

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

$$\int dz d\overline{z} \exp({-z\overline{z}}) = \int dx dy \exp({- x^2 - y^2})$$

3. The attempt at a solution
Using $z = x + iy$, 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 $dz d\overline{z} = dx dy$ ? If so, why?

2. Jul 17, 2011

### hunt_mat

What you have is $dz\wedge d\bar{z}$, compute $dz$ and $d\bar{z}$ and take their wedge product.