Removing Abs with `ComplexExpand` not working

Question

Here is what I have used for this.
From this link, Why do I get empty graph when adding 'Abs' function?
I know that I can remove Abs with ComplexExpand

Input
ComplexExpand[Sqrt[1 - Abs[x]^2]]

Output
((1 - x^2)^2)^(1/4) Cos[1/2 Arg[1 - Abs[x]^2]] + 
 I ((1 - x^2)^2)^(1/4) Sin[1/2 Arg[1 - Abs[x]^2]]

However, it works just fine when I use Plus in the Sqrt.

Input
ComplexExpand[Sqrt[1 + Abs[x]^2]]

Output
Sqrt[1 + x^2]

Is there anyone who can explain this?

Bill Watts · Accepted Answer

Maybe adding Simplify gives you what you want.
ComplexExpand[Sqrt[1 - Abs[x]^2]] // Simplify
(*Piecewise[{{I*((x^2 - 1)^2)^(1/4), Abs[x] > 1}}, ((x^2 - 1)^2)^(1/4)]*)

or
$Assumptions = -1 < x < 1

ComplexExpand[Sqrt[1 - Abs[x]^2]] // Simplify
(*Sqrt[1 - x^2]*)

The plus case is different because the value inside the Sqrt is always positive.  It is generally a good idea to add Simplify to any result ComplexExpand returns.

Removing Abs with `ComplexExpand` not working

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