Simplifying Complex Valued Expressions

Question

so I understand a little of branch cuts, but clearly not enough....

can someone please explain to me why Mathematica doesn't simplify the following expression given that the denominator is explicitly Real (and positive after squaring)

In[560]:= FullSimplify[1/Sqrt[(y z + Conjugate[y] Conjugate[z])^2] ]

Out[560]= 1/Sqrt[(y z + Conjugate[y] Conjugate[z])^2]

Probably a simple one, I know, but I really don't get it. A naive use of ComplexExpand doesn't get me anywhere either.

Is it because Mathematica thinks that 1/Abs[y z + Conjugate[y] Conjugate[z]] is more complicated?

Thanks in advance

Answer 1

expr1 = 1/Sqrt[(y z + Conjugate[y] Conjugate[z])^2];

expr2 = ComplexExpand[
   expr1 /. {y -> a + b*I, z -> c + d*I},
   TargetFunctions -> {Re, Im}] // FullSimplify

(*  1/(2*((a*c - b*d)^4)^(1/4))  *)

expr3 = expr2 /.
  {a -> Re[y], b -> Im[y], c -> Re[z], d -> Im[z]}

(*  1/(2*(((-Im[y])*Im[z] + Re[y]*Re[z])^4)^(1/4))  *)

expr1 == expr3 // FullSimplify

(*  True  *)