In my calculations on a problem of quantum optics, I faced an integral of the following form \begin{align} \frac{1}{\pi}\int e^{(\beta^* \alpha)} f(\alpha^*) d^2 \alpha \end{align} with $\alpha$ and $\beta$ complex values. May anyone help me with this problem please?
(I know that \begin{align} \frac{1}{\pi}\int e^{(\beta^* \alpha)-|\alpha|^2} f(\alpha^*) d^2 \alpha= f(\beta^*) \end{align}
(Note: The exact form of $f(\alpha)$ is $E_h (|w|^2 - \alpha ^*)$ that $E_h$ is an $h$ deformed exponential and $w$ is constant value. ))