Conjugating the following function

Question

I want to conjugate the above function. It's not working using the below code. Please help. All variables are real and greater than zero.

Refine[Conjugate[
  Exp[(I \[Pi])/4 - x^2/2 - y^2/2 + 
    Sqrt[2] E^(I \[Theta]) y \[Alpha] Sqrt[1 - \[Eta]] - 
    1/2 E^(2 I \[Theta]) \[Alpha]^2 (1 - \[Eta])]], {Element[x, 
   Reals], Element[y, Reals], Element[\[Theta], Reals], 
  Element[\[Eta], Reals], Element[\[Alpha], Reals], x > 0, y > 0}]

Edited: When I use complex expand as per the suggestion of one of the comments, I am getting a weird Arg!! Is there a way to eliminate it and look the results nicer? Complex Expand is really nice.

Answer 1

ComplexExpand[
  Conjugate[
    Exp[(I \[Pi])/4 - x^2/2 - y^2/2 + 
 Sqrt[2] E^(I \[Theta]) y \[Alpha] Sqrt[1 - \[Eta]] - 
 1/2 E^(2 I \[Theta]) \[Alpha]^2 (1 - \[Eta])]], 
TargetFunctions -> {Re, Im}] // 
  FullSimplify[#, {Element[\[Theta], Reals], Element[\[Eta], Reals], 
Element[\[Alpha], Reals], x > 0, y > 0}] &

(*   -(-1)^(3/4) E^(
 1/2 (-x^2 - y^2 + (2 E^(-I \[Theta]) y \[Alpha])/Sqrt[1/(
 2 - 2 \[Eta])] + \[Alpha]^2 (-1 + \[Eta]) Cos[
  2 \[Theta]])) (Cos[\[Alpha]^2 (-1 + \[Eta]) Cos[\[Theta]] Sin[\
\[Theta]]] - 
I Sin[\[Alpha]^2 (-1 + \[Eta]) Cos[\[Theta]] Sin[\[Theta]]])   *
)