Differential of complex conjugate

Question

I am interested change variables to the following function: \begin{equation} f(z,z^*)=\left(\frac{\partial}{\partial z^*}\right)^m \left(\frac{\partial}{\partial z}\right)^n e^{-z z^*} \end{equation} From the complex conjugates $(z,z^*)$ to the exponential variables $(w, \theta)\in\mathbb R^2$ s.t. $z=w e^{i\theta}$. I expect to find derivatives of the generalized Laguerre polynomials. I could not get to the expected result via a simple chain rule, hence I suspect that the proper change of variable is to be made under the sign of integral, and involves the Jacobian of the transformation. Could someone explain me how to solve it?