Helmuth R. Malonek

A hypercomplex derivative of monogenic functsions in and its Applications

We generalize the definition of a certain derivative of regular functions of variables from the four-dimensional real associative algebra of quaternions to monogenic functions of hypercomplex variables in Rn+1. Using this concept of derivation we look for primitives of monogenic functions in the set of monogenic functions. The results will be applied for proving a final result about the invertibility of a hypercomplex II-operator.

Power series representation for monogenic functions in based on a permutational product

This paper is a continuation of [S]. Here we propose an elementary generalization of the Weierstrass approach to holomorphic function theory into the theory of monogenic functions. The monogenic functions are generated by convergent permutational power series.

On Complete Sets of Hypercomplex Appell Polynomials

We construct a family of hypercomplex Appell polynomials and it corresponding generating function of exponential type. An extension of this approach leads to the construction of a complete set of monogenic Appell polynomials. Moreover, it is shown how the obtained Appell family is connected with complete sets of solutions of the Riesz system.

Special Monogenic Polynomials—Properties and Applications

In Clifford Analysis, several different methods have been developed for constructing monogenic functions as series with respect to properly chosen homogeneous monogenic polynomials. Almost all these methods rely on sets of orthogonal polynomials with their origin in classical (real) Harmonic Analysis in order to obtain the desired basis of homogeneous polynomials. We use a direct and elementary approach to this problem and construct a set of homogeneous polynomials involving only products of a hypercomplex variable and its hypercomplex conjugate. The obtained set is an Appell set of monogenic polynomials with respect to the hypercomplex derivative. Its intrinsic properties and some applications are presented.

Special monogenic polynomials andL2-approximation

The paper features some observations about the behaviour of special monogenic polynomials in the unit ball of ℝn which are relevant to the development of general approximation methods in the framework of Clifford Analysis. It focuses on an estimate for the best approximation of monogenic functions by a set of polynomials in Sobolev spaces. We illustrate some rather surprising orthogonality properties between symmetric homogeneous polynomials of the same degree and also an elementary method of obtaining a variety of linearly independent homogeneous polynomials of such type for the case of functions defined in ℝ3 and with values in the algebra of quaternions. The proof of their generalizations to higher dimensions in a suitable way is possible but needs more complicated combinatorical calculations which exceed the scope of this communication.

Laguerre derivative and monogenic Laguerre polynomials: An operational approach

Hypercomplex function theory generalizes the theory of holomorphic functions of one complex variable by using Clifford Algebras and provides the fundamentals of Clifford Analysis as a refinement of Harmonic Analysis in higher dimensions. We define the Laguerre derivative operator in hypercomplex context and by using operational techniques we construct generalized hypercomplex monogenic Laguerre polynomials. Moreover, Laguerre-type exponentials of order m are defined.

A note on a generalized Joukowski transformation

Abstract It is well known that the Joukowski transformation plays an important role in physical applications of conformal mappings, in particular in the study of flows around airfoils. We present, for n ≥ 2 , an n -dimensional hypercomplex analogue of the Joukowski transformation and describe in some detail the 3D case. A generalized 3D Joukowski profile, produced with Maple, is included.

A Quaternionic Beltrami-Type Equation and the Existence of Local Homeomorphic Solutions

The paper deals with a quaternionic Beltrami-type equation, which is a very natural generalization of the complex Beltrami equation to higher dimensions. Special attention is paid to the systematic use of the embedding of the set of quaternions H into C and the corresponding application of matrix singular integral operators. The proof of the existence of local homeomorphic solutions is based on a necessary and sufficient criterion, which relates the Jacobian determinant of a mapping from R into R to the quaternionic derivative of a monogenic function.

Contributions to a Geometric Function Theory in Higher Dimensions by Clifford Analysis Methods: Monogenic Functions and M-conformal mappings

We introduce the concept of M(onogenic)-conformal mappings realized by functions which are defined in an open subset of ℝ n+1 and with values in a Clifford algebra. The relation of this concept to the geometric interpretation of the hypercomplex derivative (Gurlebeck and Malonek, 1999) allows us to complete the theory of monogenic functions by providing a still missing geometric characterization of those functions. We also show that M-conformal mappings are generalizations of conformai mappings in the plane, but different from conformai mappings (in the sense of Gauss) in higher dimensions, the latter being restricted for n ≥ 2 to the set of Mobius transformations (Liouville’s Theorem).

Hypercomplex Differentiabilty and its Applications

We give a survey about the development of an elementary concept of hypercomplex differentiable A-valued functions defined in open subsets Ω of R m+1, whereby A is a Clifford algebra over the field of real numbers. Using a different from the usual one hypercomplex structure of R m+1 we get by this way a natural generalization of the Cauchy approach to monogenic functions which seems to be not possible so far. Exemplary this concept applies to transfer important properties of holomorphic functions in the plane. The results are of wide formal uniformity with the theory of functions of several complex variables.