There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Fractional Differential Equations

To say that this is the best book on the quantum theory of fields is no praise, since to my knowledge it is the only book on this subject But it is a very good and most useful book The original was written in German and appeared in 1942 This is a translation with some minor changes A few remarks have been added, concerning meson theory and nuclear forces, also footnotes referring to modern work in this field, and finally an appendix on the symmetrization of the energy momentum tensor according to Belinfante Quantum Theory of Fields Prof Gregor Wentzel Translated from the German by Charlotte Houtermans and J M Jauch Pp ix + 224, (New York and London: Interscience Publishers, Inc, 1949) 36s

Quantum Theory of Fields

Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

Fundamental theory: Holomorphic functions and domains of holomorphy Implicit functions and analytic sets The Poincare, Cousin, and Runge problems Pseudoconvex domains and pseudoconcave sets Holomorphic mappings Theory of analytic spaces: Ramified domains Analytic sets and holomorphic functions Analytic spaces Normal pseudoconvex spaces Bibliography Index.

Function theory in several complex variables

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.

A hypercomplex derivative of monogenic functsions in and its Applications

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.

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

In this paper, we consider functions defined in a star-like domain Ω⊂R n with values in the Clifford algebra C 0,n which are polymonogenic with respect to the (left) Dirac operator D = Σ n j=1 e j ∂/∂x j , i.e. they belong to the kernel of D k . We prove that any polymonogenic function f has a decomposition of the form f = f 1 + xf 2 +...+ x k-1 f k , where x = x 1 e 1 +...+x n e n and f j , j = 1,...,k, are monogenic functions. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition of polynomials. Similar results are obtained for the powers of weighted Dirac operators of the form D = |x| -α xD, ∈R/{0}.

Almansi-type theorems in Clifford analysis

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.

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Special Monogenic Polynomials—Properties and Applications

﻿ In this paper we consider three different methods for generating monogenic functions. The first one is related to Fueter's well known approach to the generation of monogenic quaternion-valued functions by means of holomorphic functions, the second one is based on the solution of hypercomplex differential equations and finally the third one is a direct series approach, based on the use of special homogeneous polynomials. We illustrate the theory by generating three different exponential functions and discuss some of their properties. Formula que se usa em preprints e artigos da nossa UI&D (acho demasiado completo): Partially supported by the R\&D unit \emph{Matem\'atica a Aplica\c\~es} (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT), co-financed by the European Community fund FEDER.

https://repositorium.sdum.uminho.pt/bitstream/1822/6002/1/81C.pdf

Helmuth R. Malonek

Papers

A hypercomplex derivative of monogenic functsions in and its Applications

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

Almansi-type theorems in Clifford analysis

Special Monogenic Polynomials—Properties and Applications

Remarks on the generation of monogenic functions