Singularities of functions of one and several bicomplex variables

Abstract

In this paper we study the singularities of holomorphic functions of bicomplex variables introduced by G. B. Price (An Introduction to Multicomplex Spaces and Functions, Dekker, New York, 1991). In particular, we use computational algebra techniques to show that even in the case of one bicomplex variable, there cannot be compact singularities. The same techniques allow us to prove a duality theorem for such functions.

The Cauchy‐Kowalewski product for bicomplex holomorphic functions

“…This section contains a short summary of the results from [7] and [8]. To begin with, we note that Remark 2.11 implies:…”

“…In this section we summarize the main properties of bicomplex numbers and of holomorphic functions of bicomplex numbers, and we refer the reader to [7], [16], [18] for further details.…”

“…More recently, we have studied some of the algebraic properties of the system of differential equations satisfied by such functions and, in the spirit of the methodology introduced and developed in [6], we have been able to deduce some new duality theorems which parallel the well-known result of Koethe-Martineau-Grothendieck for one and several complex variables. We refer the readers to [7], [8], [22] as well as to the more recent [23] for the most up-to-date description of the theory.…”

“…Finally, in [7] we were able to prove that, for any compact K in the space BC n , the following duality theorem can be proved for the sheaf H of holomorphic functions of several bicomplex variables:…”

Sato’s Hyperfunctions and Boundary Values of Monogenic Functions

“…There is another possibility to look at hyperfunctions supported in R n , in fact one may also construct hypefunctions with values in the bicomplex numbers, see [6], [7]. These are constructed as n-relative cohomology class with values in the sheaf of bicomplex holomorphic functions.…”