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Re: symbolic quaternionic analysis

• To: mathgroup at smc.vnet.net
• Subject: [mg55658] Re: symbolic quaternionic analysis
• From: danieldaniel at gmail.com (Daniel Alayon Solarz)
• Date: Fri, 1 Apr 2005 05:36:48 -0500 (EST)
• References: <d20r0d$bad$1@smc.vnet.net> <d2do5t$lit$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

> You have looked at the Quaternions package, Algebra`Quaternions` ?

yes

> 
> > 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

Thanks for this!

> 
> > 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. 

I don't think so, please run the code.


> 
> > 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]]

I am not too sure that this is faster than my implementation. Is there
any way to objectively compare speed?

> 
> 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.

This is indeed the Fueter Operator. Sudbery paper is a classic and
very recomendable. Deavours paper is **seriously flawed** and I
encourage anyone interested in this subject to read my paper:

http://www.arxiv.org/abs/math.AP/0412125

Where a genuine generalization of Cauchy-Riemann in 4D is presented.

Regards,
Daniel