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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: [mg50654] Re: [mg50617] Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]
• From: "Peter S Aptaker" <psa at laplacian.co.uk>
• Date: Wed, 15 Sep 2004 07:54:23 -0400 (EDT)
• References: <200409130619.CAA14342@smc.vnet.net> <8C2E6168-0558-11D9-A0AA-000A95B4967A@akikoz.net> <00bc01c49991$4d2d5260$4f604ed5@lap5100> <656B2636-0588-11D9-A0AA-000A95B4967A@akikoz.net>
• Sender: owner-wri-mathgroup at wolfram.com

As I said at the end of the last e-mail , my real aim is to simplify this
well known solution to a second order ODE for -1<z <1 and z >1 and w>0 . (z
is the damping ratio and w the natural frequency). The aim is to demonstrate
Mathemica with a familiar trivial problem!

dum= -((z*(-((-1 + E^((2*t*Sqrt[w^2*(-1 + z^2)])/w^2))*w*z) + (1 +
E^((2*t*Sqrt[w^2*(-1 + z^2)])/w^2))*
Sqrt[w^2*(-1 + z^2)]))/(E^((t*(w*z + Sqrt[w^2*(-1 +
z^2)]))/w^2)*R*w*Sqrt[w^2*(-1 + z^2)]))

Noting Andrzej'ssuggestion I shall take

dum2=ComplexExpand[Re[dum]];

The following take forever

Simplify[Re[dum2],w>0,z>1]

Simplify[Re[dum2],w>0,-1<z<1]



----- Original Message ----- 
From: "Andrzej Kozlowski" <andrzej at akikoz.net>
To: mathgroup at smc.vnet.net
<mathgroup at smc.vnet.net>; "Jon McLoone" <jonm at wolfram.co.uk>
Subject: [mg50654] Re: [mg50617] Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 +
eta^2]]}, eta<1]


> 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: [mg50654] 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)