Sirkka-Liisa Eriksson

Hypermonogenic Functions and their Cauchy-Type Theorems

Let Cl n be the (universal) Clifford algebra generated by e 1,…, e n satisfying e i e j +e j ,e i =−2δ ij , i,j=1,…,n. The Dirac operator in Cl n is defined by \( D = \sum\nolimits_{i = 0}^n {{e_i}\frac{\partial }{{\partial {x_i}}}} \), where e0=1. The modified Dirac operator is introduced for \(k \in \mathbb{R}\) By \( {M_K}f = Df + k\frac{{Qf}}{{{x_n}}}\), where ′ is the main involution and Qf is given by the decomposition f(x)=Pf(x)+Qf(x)e n with P f (x), Q f (x) ∈Cl nℒ1. A k+1-times continuously differentiable function f: Ω→Cl n , is called k-hypermonogenic in an open subsetΩof \( {\mathbb{R}^{n + 1}}\), if M k f (x) = 0 outside the hyperplane x n = 0. Note that 0-hypermonogenic functions are monogenic and n−1-hypermonogenic functions are hypermono-genic as defined by the authors in 15. The power function x m is hypermono-genic. The set of k-hypermonogenic functions is a right Cl n−1-module. We state a Cauchy type theorem for k-hypermonogenic functions. We also prove an integral formula for the P-part of an hypermogenic function.

Contributions to the theory of hypermonogenic functions

Let Cℓ n be the (universal) Clifford algebra generated by e 1, …, en , satisfying . The Dirac operator in Cℓ n is defined by , where e 0=1. The modified Dirac operator is introduced in (Eriksson-Bique and Leutwiler 2000, Hypermonogenic functions. In: Clifford Algebras and their Applications in Mathematical Physics, Vol. 2 (Boston: Birkhäuser), pp. 287–302) by Mf=Df+k(Q′f/x n ), where ′ denotes the main involution in Cℓ n and Qf is given by the decomposition with . A continuously differentiable function f:Ω→ Cℓ n is called hypermonogenic in an open subset Ω of , if , for all x ∈Ω. Paravector-valued hypermonogenic functions are called H-solutions, see (Leutwiler, 1992, Modified Clifford analysis, Complex Variables, 17, 153–171). The power function is an H-solution. We give a Cauchy-type formula for H-solutions. Furthermore we derive the equation for the restriction g of the hypermonogenic function f to the hyperplane xn =0. This equation has been considered by Laville, Lehman and Ramadanoff, see (2004, Analytic Cliffordian functions, Annales Academiae Scientiarium Fennicae. Mathematica, 29(2), 251–268; 1998, Holomorphic Cliffordian functions, Advances in Applied Clifford Algebras, 8(2), 323–340) and in a different context by J. Ryan, see (1990, Iterated Dirac operators in Cn , Zeitschrift fur Analysis und ihre Anwendungen, 9(5), 385–401), and others. §Dedicated to Richard Delanghe on the occasion of his 65th birthday.

Cauchy integral formula and Plemelj formula of bihypermonogenic functions in real Clifford analysis

In the first part of this article, we give the definition of bihypermonogenic functions in Clifford analysis. Using the idea of quasi-permutation, introduced by Sha Huang [Quasi-permutations and generalized regular functions in real Clifford analysis, J. Sys. Sci. and Math. Sci 18 (1998), pp. 380–384], we state an equivalent condition for bihypermonogenicity. In the second part, we discuss the Cauchy integral formula and Plemelj formula for the bihypermonogenic functions in real Clifford analysis.

On hypermonogenic functions

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.

Topics on Hyperbolic Function Theory in Geometric Algebra with a Positive Signature

In this paper we study geometric algebra valued null solutions of the equation

On Hodge-de Rham systems in hyperbolic Clifford analysis

D_{\ell}f- {k \over x_{0}}Q_{0}f=0

Fundamental solution of k-hyperbolic harmonic functions in odd spaces

on the upper half