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