what is the Jacobian for transformation from complex coordinate to real coordinate?

Question

I'm trying to figure out what is the Jacobian when you do this simple transformation:

$dzdz^* \rightarrow dxdy$

where $z=x+iy$ and $z^*=x-iy$.

Follow the formula we have $$J(x,y)= \begin{vmatrix} \frac{\partial z}{\partial x} & \frac{\partial z}{\partial y} \\ \frac{\partial z^*}{\partial x} & \frac{\partial z^*}{\partial y} \end{vmatrix} = \begin{vmatrix} 1 & i \\ 1 & -i \end{vmatrix}=-2i $$ So $dzdz^* =|J(x,y)| dxdy=2dxdy$. However, this is not what is used in literatures where they either use

$dzdz^* = dxdy$

or

$dzdz^* =2i dxdy$.

So is there a definite answer for this seemingly simple question? Thanks!

Answer 1

As differential forms: \begin{align} dz \wedge d\bar z &= (dx + i\,dy)\wedge (dx-i\,dy) \\ &= dx \wedge dx - i\, dx\wedge dy + i\,dy \wedge dx + dy \wedge dy \\ &= -2i\,dx \wedge dy \end{align}