I don't manage to make Mathematica understand that x*Conjugate[x]=Abs[x]^2.

For example, let's assume you have :


How do you make Mathematica simplify it into :



  • $\begingroup$ By most metrics the second expression is "less simple" than the second. (Hence the first expression would be prefered by Simplify, even if Mathematica was considering the proper transformations) $\endgroup$
    – TimRias
    Commented May 24, 2018 at 10:29

1 Answer 1


Try this:

expr = x*Conjugate[x]^3;

Simplify[expr /. x -> r*Exp[I*t], {r > 0, t > 0}] /. 
  t :> I*Log[Conjugate[x]/r] /. r -> Abs[x]

(*   Abs[x]^2 Conjugate[x]^2 *)

Have fun!


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