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