Conjugating symbolic expression [duplicate]

Question

I am trying to conjugate a symbolic expression, and I have explicitly stated the real terms. However, I simply can't get it to work:

Conjugate[
 ComplexExpand[
  I Cos[z] Sin[y] + Sin[z] + 
   A (Cos[z] - I Sin[y] Sin[z]), {z \[Element] Reals, 
   A \[Element] Reals, y \[Element] Reals}]]

What am I doing wrong here?

Answer 1

Just do the ComplexExpand after the Conjugate

ComplexExpand[Conjugate[I Cos[z] Sin[y] + Sin[z] + A (Cos[z] - I Sin[y] Sin[z])]]

(* A Cos[z] + Sin[z] - I (Cos[z] Sin[y] - A Sin[y] Sin[z]) *)

Answer 2

One straightforward way to take the conjugate of an expression is to replace I with -I

ComplexExpand[I Cos[z] Sin[y] + Sin[z] + A (Cos[z] - I Sin[y] Sin[z])] //. I -> (-I)

A Cos[z] + Sin[z] - I (Cos[z] Sin[y] - A Sin[y] Sin[z])

As Szabolcs points out in the comments, this solution can be problematic, so beware!

Answer 3

The simple rule I to -I is not guaranteed to work, e.g.

Exp[3 I] /. I -> -I

and ComplexConjugate might be too slow (for lengthy expressions). Therefore, I rather define an alternative function to conjugate

ClearAll[AltConjugate]
AltConjugate[x_] := ReplaceAll[FullSimplify[x], Complex[a_, b_] -> Complex[a, -b]];

This functions looks for the pattern Complex[a_, b_] and replaces it by Complex[a, -b].

@celtschk - roots might be problematic, simple functions like f[x_]=Sqrt[-x^2] can be handle by simplifying the input function, i.e. adding FullSimplify in the definition of AltConjugate. Nevertheless, this will fail for functions including more general roots, such as f[x]=Sqrt[-x^2 +I b] where both x and b are reals.

Use this carefully and always test it.

Cheers.