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Re: Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]

• To: mathgroup at smc.vnet.net
• Subject: [mg50634] Re: [mg50617] Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]
• From: Andrzej Kozlowski <andrzej at akikoz.net>
• Date: Wed, 15 Sep 2004 01:49:30 -0400 (EDT)
• References: <200409130619.CAA14342@smc.vnet.net> <8C2E6168-0558-11D9-A0AA-000A95B4967A@akikoz.net> <00bc01c49991$4d2d5260$4f604ed5@lap5100>
• Sender: owner-wri-mathgroup at wolfram.com

This is indeed most peculiar and looks like a bug. However as a 
workaround I suggest adding ComplexExpand as follows:


FullSimplify[ComplexExpand[Im[Sqrt[-1 + eta^2]]],
   -1 < eta < 1]


Sqrt[1 - eta^2]

This also works in version 4.2.

Andrzej

On 13 Sep 2004, at 21:56, Peter S Aptaker wrote:

> Sadly it does not work in M4.2 which I tend to use "for varous reasons"
>
>
> Back to M5 for now:
>
>  Simplify[{Re[Sqrt[-1+eta^2]],Im[Sqrt[-1+eta^2]]},-1<eta<1] is fine
>
> Unfortunately:
>
>
> Simplify[Im[Sqrt[-1 + eta^2]],-1<eta<1]
>
> and
>
> Simplify[{Im[Sqrt[-1+eta^2]],Im[Sqrt[-1+eta^2]]},-1<eta<1]
>
>  both leave the Im[]
>
> Thanks
> Peter
> ----- Original Message -----
>  From: "Andrzej Kozlowski" <andrzej at akikoz.net>
To: mathgroup at smc.vnet.net
> To: "peteraptaker" <psa at laplacian.co.uk>
> Cc: <mathgroup at smc.vnet.net>
> Sent: Monday, September 13, 2004 8:43 AM
> Subject: [mg50634] Re: [mg50617] Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + 
> eta^2]]}, eta<1]
>
> > *This message was transferred with a trial version of 
> CommuniGate(tm) Pro*
> > On 13 Sep 2004, at 15:19, peteraptaker wrote:
> >
> > > *This message was transferred with a trial version of 
> CommuniGate(tm)
>  > > Pro*
> > > Have I missed something - my apologies if this is answered in a FAQ
> > > I want to make the simple Re and Im parts simplify properly?
> > >
> > > test =
> > >   {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}
> > >
> > > FullSimplify[test, eta > 1]
> > > gives*{Sqrt[-1 + eta^2], 0}
> > >
> > > But
> > > FullSimplify[test, eta < 1]
> > > gives
> > > {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}
> > >
> > > Needs["Algebra`ReIm`"] does not seem to help
> > >
> > > Real numbers demonstrate what should happen:
> > > test) /. {{eta -> 0.1}, {eta -> 2}}
> > > {{0., 0.99498743710662}, {Sqrt[3], 0}}
> > >
> > >
> >
>  > There is nothing really strange here, Mathematica simply can't give 
> a
>  > single simple expression that would cover all the cases that arise. 
> So
>  > you have to split it yourself, for example:
> >
> >
> > FullSimplify[test, eta < -1]
> >
> >
> > {Sqrt[eta^2 - 1], 0}
> >
> > FullSimplify[test, eta == -1]
> >
> > {0, 0}
> >
> >
> > FullSimplify[test, -1 < eta < 1]
> >
>  > {0, Sqrt[1 - eta^2]}
> >
> >
> > FullSimplify[test, eta == 1]
> >
> >
> > {0, 0}
> >
> >
> > FullSimplify[test, 1 <= eta]
> >
> >
> > {Sqrt[eta^2 - 1], 0}
> >
> >
> > or, you can combine everything into just two cases:
> >
> > FullSimplify[test, eta $B":(B Reals && Abs[eta] < 1]
> >
> > {Re[Sqrt[eta^2 - 1]], Im[Sqrt[eta^2 - 1]]}
> >
> >
>  > FullSimplify[test, eta $B":(B Reals && Abs[eta] >= 1]
> >
>  > {Sqrt[eta^2 - 1], 0}
> >
> > In fact you do not really need FullSimplify, simple Simplify will do
>  > just as well.
> >
> >
>  > Andrzej Kozlowski
> > Chiba, Japan
> > http://www.akikoz.net/~andrzej/
> > http://www.mimuw.edu.pl/~akoz/
> >
>  >

• References:
  • Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]
    • From: psa@laplacian.co.uk (peteraptaker)