On hypermonogenic functions

Abstract

We research a function theory in higher dimensions based on the hyperbolic metric . The complex numbers are extended by the Clifford algebra Cl 0,n generated by the anti-commutating elements e i satisfying . In 1992, H. Leutwiler noticed that the power function (x 0 + x 1 e 1 + ··· + x n e n ) m is the generalized conjugate gradient of the functions . In the complex field (n = 1) this function h is harmonic in the usual sense, but in the higher dimensional case it is harmonic with respect to the Laplace–Beltrami operator with respect to the Riemannian hyperbolic metric . He started to study these type of functions, called H-solutions, that include positive and negative powers and elementary functions. Their total Clifford algebra valued generalizations, called hypermonogenic functions, are defined by H. Leutwiler and the first author in 2000. The integral formula has been proved by the first author. In this article, we present a simple way to find hyperbolic harmonic functions depending on the hyperbolic distance. We use these functions to determine a better presentation of the kernel, that is surprisingly the shifted Euclidean Cauchy kernel. We prove a power series expansion of hypermonogenic functions and present a version of the Maximum Modulus theorem.