simplifying expressions - FullSimplify on complex numbers seems inconsistent

FullSimplify[c Conjugate[c]]

returns

Abs[c]^2

while

FullSimplify[2 c Conjugate[c]]

returns

2 c Conjugate[c]

This seems strange to me. For some reason, a third entry in the Times[] precludes the simplification with Abs[]. Is this just a shortcoming of the way FullSimplify[] is implemented, or is there some good reason for this?

It's quite annoying when I have a complicated expression and some terms are real and positive definite, but this is not immediately obvious, because FullSimplify[] is failing to combine variables with their complex conjugates.

Answer

Well, FullSimplify[ ] has its own creed about what transformations should try and how to measure the complexity of its output.

You can coerce it to perform in some crazy ways, though:

exp =  2 c Conjugate[c];
SimplifyCount[p_] := First@ImageDimensions@Rasterize[p]

f[e_] := e /. Times[x___, a_, y___, Conjugate[a_], z__] :> Times[Abs[a]^2, x, y, z]
FullSimplify[exp, TransformationFunctions -> f, ComplexityFunction -> SimplifyCount]

(* 2 Abs[c]^2 *)

Comments

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