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Re: Mathematical Experiments (how to construct more functions)
 To: mathgroup at smc.vnet.net
 Subject: [mg54807] Re: Mathematical Experiments (how to construct more functions)
 From: danieldaniel at gmail.com (Daniel Alayon Solarz)
 Date: Wed, 2 Mar 2005 01:26:55 0500 (EST)
 References: <d0151r$oud$1@smc.vnet.net>
 Sender: ownerwrimathgroup at wolfram.com
I briefly explain where these come from and correct some equations
that were wrong.
Observe that in all cases we have a function acting on the spherical
coordinates. These functions are complex like and belong to the
3space. They are nontrivial solutions for 4D cauchyriemann
equations that are described with quaternions. This is my research
interest. As these solutions are pretty newborn I am still thinking
about "what" they are.
Formally, the fundamental solution is written as:
1) u + Log(Tan(v/2)i
where i means a parametrization of the sphere. To obtain more
associated solutions you can make different thinks just imagining they
are complex numbers:
i) Multiply by i, u*i  Log(Tan(v/2)), now u acts on the sphere.
ii) Take the conjugate: u  Log(Tan(v/2)i which is u + Log(Cot(v/2))i
iii) Take the inverse (conjugate divided by the square of the norm):
(1/(u² + Log(Cot(v/2)^2))(u + Log(Tan(v/2)i)
so we obtain 5 solutions of "order" 1 that can act on the sphere:
u
Log(Tan(v/2)
Log(Cot(v/2)^2))
1/(u² + Log(Cot(v/2)^2))*u
(1/(u² + Log(Cot(v/2)^2))*Log(Tan(v/2)
To obtain the we recursively constrtuct more solutions by multiplying
the fundamental solution with itself like they were complex numbers:
(u + Log(Tan(v/2))*(u + Log(Tan(v/2)) = u^2 + Log(Cot(v/2)) + i
(2*u*Log(Tan(v/2)), etc
The method that leads to these functions is described on this paper:
http://www.arxiv.org/abs/math.AP/0412125
I thank those who improved my code (I am a Mathematica neophite).
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