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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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