Hypermonogenic functions
by
Sirkka-Liisa Eriksson-Bique
Department of Mathematics, University of Joensuu, P.O. Box 111, 80101 Joensuu, Finnland

Let Cl_n be the (universal) Clifford algebra generated by e₁, ..., e_n satisfying e_i e_j + e_j e_i = -2 \delta_ij,   i, j = 1, ..., n. The Dirac operator is defined by D = \sum_i=0ⁿ e_i [(\partial)/(\partialx_i)], where e₀ = 1. The modified Dirac operator is introduces by M f = D f +(n-1) [(Q'f)/(x_n)], where ' is the main involution an Qf is given by the decomposition f(x) = Pf(x) + Q f(x) e_n with Pf(x), Qf(x) in Cl_n-1. A contiuously differentiable function f: \Omega --> Cl_n is called hypermonogenic in an open subset \Omega of Rⁿ⁺¹, if M f(x) = 0 outside the hyperplane x_n = 0. Note that the power function x^m is hypermonogenic. A function f: \Omega --> Cl_n will be called hyperbolic harmonic if [`M] M f = 0, where [`M] f(x) = [`D] f(x) - (n-1) [(Q'f(x))/(x_n)]. Note that in the real-valued case, f is hyperbolic harmonic, if and only if it satisfies the Laplace-Beltrami equation x_n \Deltaf - (n-1) [(\partialf)/(\partialx₃)] = 0 associated with the hyperbolic metric. We show that f is hypermonogenic if and only if both f and xf are hyperbolic harmonic. We also prove that a function g is monogenic in the case n odd if and only if locally g = \Delta[(n-1)/2] f for a hypermonogenic function f. We also compare hypermonogenic with the holomorphic Cliffordian functions investigated by G. Laville, I. Ramadanoff and L. Pernas.

Date received: August 8, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-54.