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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