Real part of a function

Question

How to find the real part of the following function in Mathematica? The function is $$e^{-i\psi}\langle\beta|\alpha\rangle$$. When expanded, it is given as: $$e^{-i\psi}\exp\left[\beta^{*}\alpha-\vert\alpha\vert^{2}/2-\vert\beta\vert^{2}/2\right]$$

The following is what I have written in Mathematica.

E^(-I ψ - 
 1/2 α Conjugate[α] + α Conjugate[β] - 
 1/2 β Conjugate[β])

Answer 1

Applying the advice of @Bob Hanlon, one gets the answer

expr = Exp[-I \[Psi] - 
   1/2 \[Alpha] Conjugate[\[Alpha]] + \[Alpha] Conjugate[\[Beta]] - 
   1/2 \[Beta] Conjugate[\[Beta]]]

Re[expr] // ComplexExpand[#, {\[Alpha], \[Beta]}] &

(*  E^(-(1/2) Im[\[Alpha]]^2 + Im[\[Alpha]] Im[\[Beta]] - Im[\[Beta]]^2/
  2 - Re[\[Alpha]]^2/2 + Re[\[Alpha]] Re[\[Beta]] - Re[\[Beta]]^2/2)
  Cos[\[Psi] + Im[\[Beta]] Re[\[Alpha]] - Im[\[Alpha]] Re[\[Beta]]]  *)

which is correct, but lengthily. If you need to keep it more concise you can do the following:

Step 1:

 expr2 = expr /. E^(-I \[Psi] + x_) -> E^(-I \[Psi])*Hold[E^x]

(*  E^(-I \[Psi])
  Hold[E^(-(1/
     2) \[Alpha] Conjugate[\[Alpha]] + \[Alpha] Conjugate[\[Beta]] - 
   1/2 \[Beta] Conjugate[\[Beta]])]  *)

Step 2:

ExpToTrig[expr2] /. Complex[0, y_] -> 0 // ReleaseHold

(*  E^(-(1/2) \[Alpha] Conjugate[\[Alpha]] + \[Alpha] Conjugate[\[Beta]] \
- 1/2 \[Beta] Conjugate[\[Beta]]) Cos[\[Psi]]  *)

To better visualize the result below I show it as the image:

Have fun!

Answer 2

I think I got the answer.

Re[E^(-I ψ - 
 1/2 α Conjugate[α] + α Conjugate[β] - 
 1/2 β Conjugate[β])] // ComplexExpand