MathGroup Archive: September 2004 [00270]

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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: [mg50643] 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 01:50:02 -0400 (EDT)
References: <200409130619.CAA14342@smc.vnet.net> <8C2E6168-0558-11D9-A0AA-000A95B4967A@akikoz.net> <00bc01c49991$4d2d5260$4f604ed5@lap5100> <656B2636-0588-11D9-A0AA-000A95B4967A@akikoz.net> <014601c499a8$afc7aa30$4f604ed5@lap5100> <2642EC70-05F0-11D9-A0AA-000A95B4967A@akikoz.net> <001c01c49a61$14239d00$31744ed5@lap5100>
Sender: owner-wri-mathgroup at wolfram.com

In principle, yes, but examples of platform dependent behaviour have 
come up in the past on this list. Do you mean that this time you really 
are unable to replicate the results below with Mathematica 5.0 under 
Windows (or is it Linux?)
Andrzej


On 14 Sep 2004, at 22:43, Peter S Aptaker wrote:

> Thanks
>
> Re Mac implementation, I thought the kernal was the same on all 
> platforms?
>
> Peter
> ----- Original Message ----- From: "Andrzej Kozlowski" 
To: mathgroup at smc.vnet.net
> <andrzej at akikoz.net>
> To: "Peter S Aptaker" <psa at laplacian.co.uk>
> Cc: "MathGroup" <mathgroup at smc.vnet.net>; "Jon McLoone" 
> <jonm at wolfram.co.uk>
> Sent: Tuesday, September 14, 2004 2:48 AM
> Subject: [mg50643] 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*
>> After Adam Strzebonski's response the right apprach seems to be:
>>
>>
>> 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)]));
>>
>> Refine[ComplexExpand[Re[dum]], {w > 0, z > 1}]
>>
>>
>> -(z^2/(E^((t*(z*w + Sqrt[z^2 - 1]*w))/w^2)*
>>       (R*w*Sqrt[z^2 - 1]))) +
>>    (E^((2*t*Sqrt[z^2 - 1])/w - (t*(z*w + Sqrt[z^2 - 1]*w))/
>>         w^2)*z^2)/(R*w*Sqrt[z^2 - 1]) -
>>    z/(E^((t*(z*w + Sqrt[z^2 - 1]*w))/w^2)*(R*w)) -
>>    (E^((2*t*Sqrt[z^2 - 1])/w - (t*(z*w + Sqrt[z^2 - 1]*w))/
>>         w^2)*z)/(R*w)
>>
>>
>> Refine[ComplexExpand[Re[dum]], {w > 0, -1 < z < 1}]
>>
>> Out[53]=
>> -(((1 - z^2)^(3/2)*Cos[(2*t*Sqrt[1 - z^2])/w]*
>>       Sin[(t*Sqrt[1 - z^2])/w]*z^2)/(E^((t*z)/w)*
>>       (R*w*(z^2 - 1)^2))) +
>>    ((1 - z^2)^(3/2)*Sin[(t*Sqrt[1 - z^2])/w]*z^2)/
>>     (E^((t*z)/w)*(R*w*(z^2 - 1)^2)) +
>>    ((1 - z^2)^(3/2)*Cos[(t*Sqrt[1 - z^2])/w]*
>>      Sin[(2*t*Sqrt[1 - z^2])/w]*z^2)/
>>     (E^((t*z)/w)*(R*w*(z^2 - 1)^2)) -
>>    (Cos[(t*Sqrt[1 - z^2])/w]*z)/(E^((t*z)/w)*(R*w)) -
>>    (Cos[(t*Sqrt[1 - z^2])/w]*Cos[(2*t*Sqrt[1 - z^2])/w]*z)/
>>     (E^((t*z)/w)*(R*w)) - (Sin[(t*Sqrt[1 - z^2])/w]*
>>      Sin[(2*t*Sqrt[1 - z^2])/w]*z)/(E^((t*z)/w)*(R*w))
>>
>>
>> Of course this will only work in Mathematica 5.0.
>>
>> However,  in Mathematica 5.0 on Mac OS X Simplify also works, giving
>>
>> Simplify[ComplexExpand[Re[dum]], {w > 0, z > 1}]
>>
>> -((1/(R*w*(z^2 - 1)))*((2*z*(Cosh[(t*Sqrt[z^2 - 1])/w] +
>>        Sinh[(t*Sqrt[z^2 - 1])/w])*
>>       ((z^2 - 1)*Cosh[(t*Sqrt[z^2 - 1])/w] -
>>        z*Sqrt[z^2 - 1]*Sinh[(t*Sqrt[z^2 - 1])/w]))/
>>      E^((t*(z + Sqrt[z^2 - 1]))/w)))
>>
>>
>> Simplify[ComplexExpand[Re[dum]], {w > 0, -1 < z < 1}]
>>
>>
>> -((2*z*((z^2 - 1)*Cos[(t*Sqrt[1 - z^2])/w] +
>>       z*Sqrt[1 - z^2]*Sin[(t*Sqrt[1 - z^2])/w]))/
>>     (E^((t*z)/w)*(R*w*(z^2 - 1))))
>>
>>
>>
>> Andrzej
>>
>>
>> Andrzej Kozlowski
>> Chiba, Japan
>> http://www.akikoz.net/~andrzej/
>> http://www.mimuw.edu.pl/~akoz/
>>
>>
>> On 14 Sep 2004, at 00:44, Peter S Aptaker wrote:
>>
>> > *This message was transferred with a trial version of 
>> CommuniGate(tm)
>> > Pro*
>> > 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
>> > To: "Peter S Aptaker" <psa at laplacian.co.uk>
>> > Cc: "Adam Strzebonski" <adams at wolfram.com>; "MathGroup"
>> > <mathgroup at smc.vnet.net>; "Jon McLoone" <jonm at wolfram.co.uk>
>> > Sent: Monday, September 13, 2004 2:25 PM
>> > Subject: [mg50643] 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*
>> >> 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: [mg50643] 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)

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