suivant: Bloch constant for -holomorphic monter: A Bloch Constant for précédent: Introduction

# Preliminaries

Here, we introduce some of the basic results of the theory of bicomplex numbers. First, we define bicomplex numbers as follows: where . The norm used on is the Euclidean norm (also noted ) of . We remark that we can write a bicomplex number as where . Thus, can be veiwed as the complexification of and a bicomplex number can be seen as an element of . It is easy to see [13] that is a commutative unitary ring with the following characterization for the noninvertible elements:

Proposition 1   Let . Then is noninvertible iff and or and iff .

It is also possible to define differentiability of a function at a point of [13]:

Definition 2   Let be an open set of and . Then, is said to be -differentiable at with derivative equal to if

We will also say that the function is -holomorphic on an open set U iff is -differentiable at each point of U.

As we saw, a bicomplex number can be seen as an element of , so a function of can be see as a mapping of . Here we have a characterization of such mappings:

Theorem 3   Let U be an open set and such that . Let also . Then is -holomorphic on U iff:

and,

Moreover, and is invertible iff .

This theorem can be obtained from results in [13] and [14]. Moreover, by the Hartogs theorem [19], it is possible to show that " can be dropped from the hypotheses. Now, it is natural to define for the following class of mappings introduced in [14]:

Definition 3   The class of -holomorphic mappings on a open set is defined as follows:

It is the subclass of holomorphic mappings of satisfying the complexified Cauchy-Riemann equations.

In [14], bicomplex numbers were called tetranumbers and was denoted by . In this article, we will use this notation when a definition can be written independently of the theory of bicomplex numbers . We remark that iff is -holomorphic on U. It is also important to know that every bicomplex number has the following unique idempotent representation:

where and . This representation is very useful because: addition, multiplication and division can be done term-by-term. Also, an element will be noninvertible iff or . The notion of holomorphicity can also be see with this kind of notation. For this we need to define the functions as and . Also, we need the following definition:

Definition 4   We say that is a -cartesian set determined by and if .

In [13] it is shown that if and are domains of then is also a domain of . Now, it is possible to state the following striking theorems [13]:

Theorem 4   If and are holomorphic functions of on the domains and respectively, then the function defined as

is -holomorphic on the domain

Theorem 5   Let be a domain in , and let be a -holomorphic function on X. Then there exist holomorphic functions and with and , such that:

We note here that and will also be domains of .

Finally we give some concrete simple examples of entire -holomorphic mappings which come from the bicomplex theory. First, the exponential mapping defined as:

with and . Secondly, the polynomials of degree n, with bicomplex multiplication, defined as:

with for and . We note that the derivative will be what we expect for polynomials.

suivant: Bloch constant for -holomorphic monter: A Bloch Constant for précédent: Introduction
Dominic Rochon
2000-07-26