How can Complex numbers be written in this way?

Question

In quantum optics, we assume that the eigenvalues of coherent states are complex numbers. So, when we determine the overlap of two different coherent states we have to deal with complex numbers all the time. In an academic book, I encountered an equation and I want to know how complex numbers can be expressed in this way:

Assume that $\alpha$ and $\beta$ are complex numbers, we have:

$e^{\alpha^*\beta-\alpha\beta^*}$=$e^{2i{\Im}(\alpha^*\beta)}$

Can somebody explain for me, how $\alpha^*\beta-\alpha\beta^* = 2i{\Im}(\alpha^*\beta)$? Are there any equivalent expressions? Can we say that it is also equal to $2i{\Im}(\alpha\beta^*)$ or $-2i{\Im}(\alpha^*\beta)$?

Answer 1

Write $\alpha^\ast\beta=a+bi$ with $a,\,b\in\Bbb R$ so $\alpha\beta^\ast=(\alpha^\ast\beta)^\ast=a-bi$ and$$\alpha^\ast\beta-\alpha\beta^\ast=2bi=2i\Im(\alpha^\ast\beta)=-2i\Im(\alpha\beta^\ast).$$

Answer 2

$$\alpha^*\beta-\alpha\beta^*=\alpha^*\beta-(\alpha^*\beta)^*=2i\Im(\alpha^*\beta),$$ simply.

Answer 3

It is true for any complex number $z$ that $z - z^* = 2i\operatorname{Im}(z)$. This is easy to confirm, both geometrically in the plane, and algebraically by setting $z = x + yi$.