Hadamard-like complex variable substitution \begin{align}
\frac\pi a &= \int_{-\infty}^\infty dxdye^{-a(x^2+y^2)}\\
\tag{1}&= \int_{-\infty}^\infty dxdye^{-a(x+iy)(x-iy)}
\end{align}
So far so good. Now introduce a complex variable $z$ and its conjugate $z^*$ such that
$$
x = \frac{z+z^*}{2}, y=\frac{z-z^*}{2i}
$$
I suppose that means $z=x+iy$ and $z=x-iy$.
According to my professor, this means that the Gauss integral $(1)$ becomes
\begin{align}
\tag{2}\frac\pi a &= \frac{1}{2i}\int dzdz^*e^{-azz^*}
\end{align}
I don't understand how to substitute the variables, and from where to where this integral runs.
I tried it like this:
$$
dx=\frac 12(dz+dz^*), dy=\frac 1{2i}(dz-dz^*)\\
dxdy = \frac{1}{4i}(dz^2-{dz^*}^2)
$$


*

*How can I show that the last term equals $\frac{1}{2i}dzdz^*$?

*What are the limits of the integral $(2)$?


The substitution you made yields the Jacobian
$$\begin{vmatrix}\cfrac12&\;\;\cfrac12\\\cfrac1{2i}&-\cfrac1{2i}\end{vmatrix}=-\frac1{2i}\implies dxdy=-\frac1{2i}dzdz^*$$
The minus sign is not that relevant here as without a specific integration path, and thus some orientation, it rather is moot.
I can't tell anything about the path taken, but it would seem to be something in $\;\Bbb C\times\Bbb C\;$ . I really don't know.
