# Rewrite $x\sqrt{1-y^2/x^2} = \text{sgn}(x)\sqrt{x^2-y^2}$

I'm trying to coax Mathematica into rewriting $$x\sqrt{1-y^2/x^2} = \text{sgn}(x)\sqrt{x^2-y^2}$$ for $$x$$ and $$y$$ real. I have tried all combinations of FullSimplify and Refine with the appropriate assumptions. Mathematica refuses to cooperate.

With the additional assumption $$x>0$$, I can get Mathematica to simplify to $$\sqrt{x^2-y^2}$$, but I don't want to assume this. I want the general result with the $$\text{sgn}(x)$$.

• Sometimes, it is actually hard to tell if one form is simpler than another. But if *Simplify etc. refuse to carry out further transformations, that means, Mathematica determines, according to its own built-in principles, that the "simplest" form has already been reached. – Αλέξανδρος Ζεγγ Nov 18 '18 at 14:13

For this particular expression you can use

ComplexExpand[Re[x Sqrt[1 - y^2/x^2]], {x, y}] //
FullSimplify[#,
FunctionDomain[x Sqrt[1 - y^2/x^2], {x, y}, Reals]] & // ExpandAll

(*Sqrt[x^2 - y^2] Sign[x]*)


You can also use TargetFunctions with Sign or RealSign as an option to ComplexExpand for more complicated expressions.

It is not clear, why do you need to make such a replacement by Mathematica, when it is much faster to straightforwardly type the desired expression.

However, if you are insisting, try this

expr = x*Sqrt[1 - y^2/x^2];
expr1 = expr /. x -> Sign[z]*Abs[z]

(*  Abs[z] Sqrt[1 - y^2/(Abs[z]^2 Sign[z]^2)] Sign[z]  *)


and then

(expr1 // Simplify[#, z ∈ Reals && z != 0] &) /. z -> x

(*  Sqrt[x^2 - y^2] Sign[x]  *)


Done. Have fun!

• I just gave a simplified example for the purposes of asking the question. I don't need to simplify this expression in particular. I need to be able to give Mathematica an expression of unknown form and allow it to perform the simplification using $\text{sgn}$, if it's possible. – Donjon Oct 22 '18 at 8:14
• So, I gave you an example. There are, however, several methods. One chooses the best one according to the expression. – Alexei Boulbitch Oct 23 '18 at 18:29