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

For example, let's assume you have :

x*Conjugate[x]^3

How do you make Mathematica simplify it into :

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

?

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  • $\begingroup$ 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) $\endgroup$ – mmeent May 24 '18 at 10:29
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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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