Complex Analysis with Mathematica, Simplify, Absolute value

Question

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 

?

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!