# identity for integration of analytic function

I came across an identity for integration of analytic function , and I don't know how to prove it. Please help me with it, thank you. The statement goes as following

For any analytic function $$f(\alpha)$$ $$$$f(\beta) = \frac{1}{2\pi}\int f(\alpha)\exp{\{-|\beta-\alpha|^2/2\}}d^2\alpha$$$$ in which $$d^2\alpha=d\mathrm{Re}\{\alpha\}d\mathrm{Im}\{\alpha\}$$

This identity follows from Gauss mean value theorem: If $$f:\Bbb C\to \Bbb C$$ is entire (with moderate growth), we have \begin{align*} \int_{\Bbb R^2}f(\alpha)e^{-|\beta-\alpha|^2/2}d\alpha&=\int_{\Bbb R^2}f(\beta+\alpha)e^{-|\alpha|^2/2}d\alpha\\&=\int_{0}^{2\pi}\int_0^\infty f(\beta+re^{i\theta})e^{-\frac{r^2}2}r drd \theta\tag{*}\\&=\int_0^\infty\left(\int_{0}^{2\pi} f(\beta+re^{i\theta})d \theta\right)e^{-\frac{r^2}2}r dr\\&=2\pi f(\beta)\cdot \int_0^\infty e^{-\frac{r^2}2}r dr\tag{**}\\&=2\pi f(\beta)\left[-e^{-\frac{r^2}2}\right]^\infty_0=2\pi f(\beta). \end{align*} $$(*)$$ : polar coordinate change.
$$(**)$$ : mean value theorem is used.