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MathGroup Archive 2005

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Re: positive square root

  • To: mathgroup at smc.vnet.net
  • Subject: [mg58700] Re: positive square root
  • From: Peter Pein <petsie at dordos.net>
  • Date: Fri, 15 Jul 2005 03:02:15 -0400 (EDT)
  • References: <db57ne$4no$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

paulvonhippel at yahoo schrieb:
> I work in a world where the square root is always a positive number.
> But Mathematica allows for the possibility of negative square roots.
> Two questions arise:
> 
> (1) Is there a way to tell Mathematica that I'm only interested in
> positive square roots?
> 
> (2) My current solution is to use, e.g., Abs[Sqrt[z]]. But when
> Mathematica echoes this, it puts the Abs function *under* the radical,
> so it looks like Sqrt[Abs[z]]. Is this a bug in the display?
> 
Hi Paul,

I don't think so. It is just in general easier, to evaluate Sqrt[Abs[z]]
than Abs[Sqrt[z]]:

In[1]:=
Assuming[(x | y) \[Element] Reals,
  FullSimplify[
   {Abs[ComplexExpand[Sqrt[x + I*y]]],(*sqrt first*)
    ComplexExpand[Sqrt[Abs[x + I*y]]]}]](*abs first*)
Out[1]=
{(x^2 + y^2)^(1/4)*
  Abs[Cos[(1/2)*ArcTan[x, y]] + I*Sin[(1/2)*ArcTan[x, y]]],
x^2 + y^2)^(1/4)}

These are of course equivalent:

In[2]:=
SameQ @@ Simplify[ComplexExpand[%]]
Out[2]=
True

-- 
Peter Pein
Berlin
http://people.freenet.de/Peter_Berlin/


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