In the book, "Quantum Continuous Variables (A Primer of Theoretical Methods)" by Alessio Serafini, on page 70, he defines an $s$-ordered characteristic function to be: $$ \chi_s(\alpha)=\operatorname{Tr}\left[\hat{D}_\alpha \varrho\right] \mathrm{e}^{\frac{s}{2}|\alpha|^2}. $$ Then (on page 73), he introduces the concept of quasi-probability distributions and defines an "$s$-ordered quasi-probability distribution" $W_s(\alpha)$ to be the 'complex Fourier transform' of $\chi_s(\alpha)$, which turns out to be: $$ W_s(\alpha)=\frac{1}{\pi^2} \int_{\mathbb{C}} \mathrm{d}^2 \beta \mathrm{e}^{\left(\alpha \beta^*-\alpha^* \beta\right)} \chi_s(\beta). $$ Here, $\alpha$ and $\beta$ are eigenvalues of annihilation operator and take the form of $\alpha = \frac{x+ip}{\sqrt{2}}$ (same goes for $\beta$).
I don't understand how did the term $\mathrm{e}^{\left(\alpha \beta^*-\alpha^* \beta\right)}$ in $W_s(\alpha)$ come about. As far as I can think, the only time I've seen that sort of term is in the following property of the displacement operator (Weyl Operator): $$ \hat{D}_\alpha \hat{D}_\beta=\mathrm{e}^{\frac{1}{2}\left(\alpha \beta^*-\alpha^* \beta\right)} \hat{D}_{\alpha+\beta}. $$
Please do answer knowing that I just know the basics of Fourier transform that one learns in their first 'Mathematical Methods in Physics' course. Thank you!
