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MathGroup Archive 2005

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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: owner-wri-mathgroup 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
3-space. They are non-trivial solutions for 4D cauchy-riemann
equations that are described with quaternions. This is my research
interest. As these solutions are pretty new-born 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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