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Re: Mathematical Experiments (how to construct more functions)
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).