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MathGroup Archive 2004

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Re: Exact real solutions of cubic equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg46963] Re: Exact real solutions of cubic equations
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Thu, 18 Mar 2004 01:24:47 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

In article <c3906l$gce$1 at smc.vnet.net>,
 Paul Abbott <paul at physics.uwa.edu.au> wrote:

> > Does anyone know of a package that can simplify expressions
> > with complex numbers? 
> 
> Physics 52 (3), 269 (1984)), a cubic of the form  x^3 - a x + b has 
> three real roots when a > 0 and 27 b^2 < 4 a^3. In this case, the roots 
> can be written as 
> 
>   r[n_] = 2 Sqrt[a/3] Cos[Pi (2n - 1)/3 + ArcCos[(b/2)/(a/3)^(3/2)]/3]
> 
> where n = 1,2,3. This is easily verified:

Actually, I wrote (as the email to myself verifies):

  From the general theory of cubic equations (see American Journal of 
  Physics 52 (3), 269 (1984)), a cubic of the form  x^3 - a x + b has 
  three real roots when a > 0 and 27 b^2 < 4 a^3. In this case, the 
  roots can be written as 

    r[n_] = 2 Sqrt[a/3] Cos[Pi (2n - 1)/3 + ArcCos[(b/2)/(a/3)^(3/2)]/3]

  where n = 1,2,3. This is easily verified:

but somehow, the mathgroup newsgroup system truncated the first line of 
my response. This is not the first time that this has happened. It seems 
to occur if there is a spurious > as the first character on a line.
[This problem has been fixed - moderator]

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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