[Date Index]
[Thread Index]
[Author Index]
Re: symbolic quaternionic analysis
*To*: mathgroup at smc.vnet.net
*Subject*: [mg55561] Re: symbolic quaternionic analysis
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Wed, 30 Mar 2005 03:20:56 -0500 (EST)
*Organization*: The University of Western Australia
*References*: <d20r0d$bad$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <d20r0d$bad$1 at smc.vnet.net>,
danieldaniel at gmail.com (Daniel Alayon Solarz) wrote:
> At the end of this message is my code to handle what the title refers to.
>
> I have two questions,
>
> 1) is there any other?
You have looked at the Quaternions package, Algebra`Quaternions` ?
> 2) I am interested in functional analysis, is Mathematica capable of handle
> that?
It is extensible. For example, have a look at
http://physics.uwa.edu.au/pub/Mathematica/MathGroup/Quaternions.nb
> p[t_, x_, y_, z_] := {t, {x, y, z}}
> m[p[t1_, x1_, y1_, z1_],
> p[t2_, x2_, y2_, z2_]] := {t1*t2 - Dot[{x1, y1, z1}, {x2, y2, z2}],
> t1*{x2, y2, z2} + t2*{x1, y1, z1} + Cross[{x1, y1, z1}, {x2, y2, z2}]}
The result of the multiplication should itself be a quaternion. You are
returning just a list. Also you could work with the vector part
{x, y, z} as a single object.
> Pwr[p[t_, x_, y_, z_], 0] := { 1, {0, 0, 0}}
> Pwr[p[t_, x_, y_, z_], 1] := p[t, x, y, z]
> Pwr[p[t_, x_, y_, z_], n_] := m[Pwr[p[t, x, y, z], n - 1], p[t, x, y, z]]
Note that de Moivre's identity holds for quaternions. The n-th power of
Quaternion[s, v]
where v = {x,y,z} is
Quaternion[Re[(s + I r)^n], v/r Im[(s + I r)^n]]
where r = Sqrt[v.v]. This result can be used, formally at least, to
implement analytic functions with real coefficients.
> RFueter[{a_, {b_, c_, d_}}] :=
> m[D[{a, {b, c, d}}, t], p[1, 0, 0, 0]] +
> m[D[{a, {b, c, d}}, x], p[0, 1, 0, 0]] +
> m[D[{a, {b, c, d}}, y], p[0, 0, 1, 0]] +
> m[D[{a, {b, c, d}}, z], p[0, 0, 0, 1]]
> LFueter[{a_, {b_, c_, d_}}] :=
> m[p[1, 0, 0, 0], D[{a, {b, c, d}}, t]] +
> m[p[0, 1, 0, 0], D[{a, {b, c, d}}, x]] +
> m[p[0, 0, 1, 0], D[{a, {b, c, d}}, y]] +
> m[p[0, 0, 0, 1], D[{a, {b, c, d}}, z]]
I am aware of the Cauchy-Riemann-Fueter equations but this does not look
like a correct implementation of them to me. I also have copies of
papers by Deavours, Sudbury, and Sweetster that may be of interest to
you.
Cheers,
Paul
--
Paul Abbott Phone: +61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 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
Prev by Date:
**Mathematica hangs when solving sys. of equations w/ certain parameters**
Next by Date:
**Re: Much faster ConvexHull implementation**
Previous by thread:
**Re: symbolic quaternionic analysis**
Next by thread:
**MapAt changes its response towards different expressions**
| |