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).