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MathGroup Archive 2006

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Re: Simplify question

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71725] Re: [mg71655] Simplify question
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 27 Nov 2006 04:05:09 -0500 (EST)
  • References: <200611260848.DAA14451@smc.vnet.net>

On 26 Nov 2006, at 17:48, dimitris wrote:

> The following list of expressions was obtained by following the steps
> of Tartaglia's solution of the cubic equation with Mathematica.
>
> lstcub = {{((-1)^(1/3)*Q)/(R - Sqrt[Q^3 + R^2])^(1/3) + (-1)^(2/3)* 
> (R -
> Sqrt[Q^3 + R^2])^(1/3),
>      -(Q/(R - Sqrt[Q^3 + R^2])^(1/3)) + (R - Sqrt[Q^3 + R^2])^(1/3),
> -(((-1)^(2/3)*Q)/(R - Sqrt[Q^3 + R^2])^(1/3)) -
>       (-1)^(1/3)*(R - Sqrt[Q^3 + R^2])^(1/3)}, {-(Q/(R + Sqrt[Q^3 +
> R^2])^(1/3)) + (R + Sqrt[Q^3 + R^2])^(1/3),
>      ((-1)^(1/3)*Q)/(R + Sqrt[Q^3 + R^2])^(1/3) + (-1)^(2/3)*(R +
> Sqrt[Q^3 + R^2])^(1/3),
>      -(((-1)^(2/3)*Q)/(R + Sqrt[Q^3 + R^2])^(1/3)) - (-1)^(1/3)*(R +
> Sqrt[Q^3 + R^2])^(1/3)}};
>
> TableForm[%]//TraditionalForm
>
> Although the solution of the reduced cubic equation can be obtained
> making a tricky observation
>
> (see e.g. Leonard E. (Leonard Eugene) Dickson b. 1874. (page 32) ,
> Elementary theory of equations.  1914.
> available online at
> http://mathbooks.library.cornell.edu:8085/Dienst? 
> verb=Display&protocol=CGM&ver=1.0&identifier=cul.math/01460001"
>
> target=_blank
>> http://mathbooks.library.cornell.edu:8085/Dienst? 
>> verb=Display&protocol=CGM&ver=1.0&identifier=cul.math/01460001)
>
> so what I ask don't play any important role in the solution procedure,
> I wonder if Mathematica can verify the equalities
>
> MapThread[Equal, lstcub]
> {((-1)^(1/3)*Q)/(R - Sqrt[Q^3 + R^2])^(1/3) + (-1)^(2/3)*(R - Sqrt[Q^3
> + R^2])^(1/3) ==
>    -(Q/(R + Sqrt[Q^3 + R^2])^(1/3)) + (R + Sqrt[Q^3 + R^2])^(1/3),
>   -(Q/(R - Sqrt[Q^3 + R^2])^(1/3)) + (R - Sqrt[Q^3 + R^2])^(1/3) ==
> ((-1)^(1/3)*Q)/(R + Sqrt[Q^3 + R^2])^(1/3) +
>     (-1)^(2/3)*(R + Sqrt[Q^3 + R^2])^(1/3), -(((-1)^(2/3)*Q)/(R -
> Sqrt[Q^3 + R^2])^(1/3)) -
>     (-1)^(1/3)*(R - Sqrt[Q^3 + R^2])^(1/3) == -(((-1)^(2/3)*Q)/(R +
> Sqrt[Q^3 + R^2])^(1/3)) -
>     (-1)^(1/3)*(R + Sqrt[Q^3 + R^2])^(1/3)}
>
> which can be justified in view of the results
>
> Table[lstcub /. {R -> Random[], Q -> Random[]}, {5}] // Chop
>
> Thanks a lot for any response.
>
> Dimitris
>

It clearly can't because:


In[1]:=
FindInstance[((-1)^(1/3)*Q)/(R - Sqrt[Q^3 + R^2])^(1/3) +
     (-1)^(2/3)*(R - Sqrt[Q^3 + R^2])^(1/3) != -(Q/(R + Sqrt[Q^3 +  
R^2])^(1/3)) +
     (R + Sqrt[Q^3 + R^2])^(1/3), {R, Q}]

Out[1]=
{{R -> -7 + (15*I)/2, Q -> 18/5 + (34*I)/5}}

etc,

Andrzej Kozlowski
Tokyo, Japan




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