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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: [mg50641] 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 01:49:52 -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>
• Sender: owner-wri-mathgroup at wolfram.com

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: [mg50641] 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: [mg50641] 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)