# Why doesn't Conjugate[] conjugate exponentials inside other functions?

Why does something like Conjugate[Sqrt[Exp[I x]]] give output Conjugate[Sqrt[E^(I x)]] instead of Sqrt[E^(-I x)]?

In some places I get outputs such as: Conjugate[0.752066 E^((3 I \[Pi])/2 - (0. + 3.46164 I) x - x^2/2)]. Why does this happen?

• Use ESC+e+e+ESC for e and ESC+i+i+ESC for i. Commented Jul 9 at 21:43
• Or use Exp[I] Commented Jul 9 at 21:57
• Use Exp[ ] for exponential function and (capital) I for the imaginary unit. Conjugate[Exp[I]] correctly returns Exp[-I] as expected. Commented Jul 10 at 0:42
• I use ESC+e+e+ESC and the same for i. I only wrote like that for simplicity. Exp[I] works on its own, but not inside other functions. But still I was not aware of it thank you for pointing it out. Commented Jul 10 at 4:56
• @A.Kato you are right. I will edit my question accordingly. Commented Jul 10 at 11:01

Mathematically speaking, conjugation of a function $$\bar{f}(z)$$ is not necessarily same as the same function evaluated at the conjugate argument $$f(\bar{z})$$. As an example, if $$f(z)=z+i$$, clearly $$\bar{f}(z)\ne f(\bar{z})$$. Therefore, Mathematica cannot replace $$\bar{e}(z)$$ with $$e(\bar{z})$$ if $$e$$ is undefined, which is why Conjugate[e^i] does not evaluate to something simpler

In case of Exp[x], Mathematica does realize that exponential is a function which obeys Schwarz Reflection Principle, hence evaluates Conjugate[Exp[f[x]]] to Exp[Conjugate[f[x]]].

• There are outputs in my code that are of the form Conjugate[E^(0+0.37294 I)]. Commented Jul 10 at 5:46
• @EFETÜRBEDAR Why didn't you add this concrete example Conjugate[E^(0+0.37294 I)] in your question? Commented Jul 10 at 8:39
• @UlrichNeumann I really should have. I wiill fix my question. Commented Jul 10 at 11:03

Why does something like Conjugate[Sqrt[Exp[I x]]] give output Conjugate[Sqrt[E^(I x)]] instead of Sqrt[E^(-I x)]?

Because the two are not equivalent, for example:

{Conjugate[Sqrt[Exp[I x]]], Sqrt[E^(-I x)]} /. x -> 1 + I
FunctionExpand[%]
N[%%]


{Conjugate[Sqrt[E^(-1 + I)]], Sqrt[E^(1 - I)]}

{E^(-(1/2) - I/2), E^(1/2 - I/2)}

{0.532281 - 0.290786 I, 1.44689 - 0.790439 I}

• Ok but they are the same when x is not complex, and even when it is shouldn't it output Sqrt[Exp[-I Conjugate[x]]] ? Also even after putting in values for x, I am left with expressions that have Conjugate[] in them. Commented Jul 10 at 18:49
• @EFETÜRBEDAR "Also even after putting in values for x, I am left with expressions that have Conjugate[] in them." Use FunctionExpand afterwards like I did in my answer. Then maybe use also ComplexExpand if needed. Commented Jul 10 at 20:28

Do any of these help?

Complex $$x$$:

ComplexExpand[Conjugate[Sqrt[E^(I  x)]], x] // FullSimplify
(*  E^(-(Im[x]/2))/Sqrt[E^(I Re[x])]  *)


Real $$x$$:

ComplexExpand[Conjugate[Sqrt[E^(I  x)]]] // FullSimplify
(*  1/Sqrt[E^(I x)]  *)


Complex $$x$$:

PowerExpand[Conjugate[Sqrt[E^(I  x)]], Assumptions -> True]
(*  E^(-(1/2) I Conjugate[x] + I π Floor[1/2 - Re[x]/(2 π)])  *)


Real $$x$$:

PowerExpand[Conjugate[Sqrt[E^(I  x)]],
Assumptions -> x \[Element] Reals]
(*  E^(-((I x)/2) + I π Floor[1/2 - x/(2 π)])  *)