On Clifford analysis for holomorphic mappings

Abstract

In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n and m, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the CauchyRiemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention is mostly restricted to the case of low dimensions where it is possible to identify new and interesting properties and to single out the special role played by bicomplex analysis.

Bicomplex holomorphic functional calculus

“…In this section we summarize the main properties of bicomplex numbers and of holomorphic functions of bicomplex numbers, and we refer the reader to , , , , , for further details.…”

Generalization of the Gauss--Lucas theorem for bicomplex polynomials

“…Corrado Segre published a paper [13] in 1892, in which he studied an infinite set of algebra whose elements he called bicomplex numbers. The work of Segre remained unnoticed for almost a century, but recently mathematicians have started taking interest in the subject and a new theory of special functions has started coming up [6,9]. In this paper, we introduce the mathematical tools necessary to investigate the Gauss-Lucas theorem for bicomplex polynomials.…”

Topological Bicomplex Modules

“…For more details on real-valued norm and hyperbolic-valued (D-valued) norm see, [1,Section 4.1,4.2]. For further details on bicomplex analysis, we refer the reader to [1], [8], [9], [14], [15], [16], [17], [18], [20], [22], [23] and references therein.…”