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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: [mg50625] 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 01:49:17 -0400 (EDT)
• References: <200409130619.CAA14342@smc.vnet.net>
• Reply-to: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Sender: owner-wri-mathgroup at wolfram.com

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)