I want to make a Taylor expansion of a given real function $f$ which depends on two complex variables.
In a general case I have :
$$f(\psi, \psi^*)=\sum_{p,q} \frac{\psi^q {\psi^{*}}^p}{p! q!} \frac{\partial^{p+q} f}{\partial \psi^p \partial {\psi^*}^q}_{| \psi=\psi^*=0}$$
But my function $f$ is real, so I have conditions on $p$ and $q$ in my sum.
I would say that $p=q$ to have $\psi^q {\psi^{*}}^p$ real, but how to be sure that the derivative of the function will also be real ?
In fact I'm looking for the general condition on $p$ and $q$ to be sure that my developpment will be real.
For instance I don't know how to be sure that the derivatives of $f$ will be real.