# Conjugating the following function

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.

• 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])]] // Refine // ComplexExpand ? Dec 16, 2020 at 3:53
• But I would like to let Mathematica know that all my variables are real. Dec 16, 2020 at 4:37

## 1 Answer

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]]])   *
)

• Brilliant!!! Working Dec 16, 2020 at 4:52