# Complex Laplacian and Derivatives of Bicomplex Functions

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DOI: 10.1007/s11785-013-0284-8

- Cite this article as:
- Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C. et al. Complex Anal. Oper. Theory (2013) 7: 1675. doi:10.1007/s11785-013-0284-8

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## Abstract

In this paper we study in detail the theory of bicomplex holomorphy, in the context of the several ways in which bicomplex numbers can be considered. In particular we will show how the notions of bicomplex derivability and bicomplex holomorphy can be interpreted in these different ways, and the consequences that can be derived.

### Keywords

Bicomplex derivabilityBicomplex differentiabilityBicomplex holomorphic functionsComplex and hyperbolic Laplacians### Mathematics Subject Classification (2010)

30G3532A3032A10## 1 Introduction

In the last several years, the theory of bicomplex numbers has enjoyed a renewed interest, mostly because of [2–4, 9, 11], and the theory of functions which are bicomplex holomorphic (or \(\mathbb{B }\mathbb{C }\)-holomorphic as we will say in this paper) has been developed along many interesting directions, including extensions to several variables [16], extension to the case of multicomplex numbers [17], extension to pseudoanalytic functions theory [1, 12]; elements of the geometric function theory for bicomplex functions can be found in [5]. The traditional text which lays out the foundations for the study of bicomplex numbers and of the functions defined on them is the monograph of Price [10], where some historical notes on these numbers are also given, while an elementary introduction of the first examples of \(\mathbb{B }\mathbb{C }\)-holomorphic functions is given in [8]. In no way this brief review pretends to cover all the sources about bicomplex numbers and their functions.

A careful reading of the monograph of Price reveals many important intuitions, that have never been sufficiently explored, but if a critical commentary can be made, one should say that the many detailed computations which are carried out in [10], often appear to lack motivation, and there is no analysis of why such calculations end up being so successful. In a word, the theory of \(\mathbb{B }\mathbb{C }\)-holomorphy is presented almost as isolated from the theory of differentiability for functions in four real variables, and from the theory of holomorphy for functions of two complex variables.

The major contribution we would like to offer with this paper is a detailed analysis of the theory of \(\mathbb{B }\mathbb{C }\)-holomorphy, in the context of the several alternative ways in which bicomplex numbers can be considered. Let us be a bit more specific. To begin with, we consider the usual field of complex numbers \(\mathbb{C }(\mathbf{i})=\{x+\mathbf{i}y \,\,: \,\, x,y\in \mathbb{R }\}.\) Note that we have made explicit the fact that we are using the imaginary unit \(\mathbf{i}\), since a second imaginary unit \(\mathbf{j}\) is now necessary to define bicomplex numbers. Indeed, we define the ring of bicomplex numbers, and we denote it by \(\mathbb{B }\mathbb{C }\), to be the ring of numbers of the form \(Z=z_1+\mathbf{j}z_2\), where both \(z_1=x_1+\mathbf{i}y_1\) and \(z_2=x_2+\mathbf{i}y_2\) are complex numbers in \(\mathbb{C }(\mathbf{i}).\) So, in a sense, a bicomplex number can be identified with a pair of complex numbers in \(\mathbb{C }(\mathbf{i}).\) However, a simple rearrangement of the terms shows that the same number \(Z\) can also be seen as a pair of complex numbers in \(\mathbb{C }(\mathbf{j})\). Even more unexpected, if we denote by \(\mathbf{k}\) the hyperbolic unit \(\mathbf{i}\mathbf{j}\) (by hyperbolic unit we simply mean that \(\mathbf{k}^2=1\)), and if we denote by \(\mathbb{D }\) the ring of hyperbolic numbers defined by \(\mathbb{D }:= \{ x + \mathbf{k}y \, \big | \, x, y \in \mathbb{R }\},\) then any bicomplex number can also be seen as a pair of hyperbolic numbers. The situation is actually even more complicated than it appears from this quick discussion, as it will be shown in detail in Sect. 2.

These three different structures (and as we mentioned, they are not the only ones we will be working with!) make it possible to interpret every definition in several different ways. It is our goal in this paper to exploit these multiple representations, and to show how the notions of \(\mathbb{B }\mathbb{C }\)-derivability and of \(\mathbb{B }\mathbb{C }\)-holomorphy can be reinterpreted in these settings, and what are the consequences that one may infer. As the reader will easily see, this approach will provide many interesting surprises and novelties, that were at a minimum not explicit (and in many cases not even contemplated) in [10].

The paper is fully self-contained and consists of six sections in addition to the introduction. Section 2 is dedicated to the definition of the ring of bicomplex numbers, and to the introduction of the various structures to which we have hinted above. We show, for example, that one can write bicomplex numbers in at least eight different ways, and that each different representation conveys a different flavor and has different consequences. In addition we recall that bicomplex numbers admit several different notions of conjugation, and correspondingly several different notions of modulus. We then come to the most important point in the theory of bicomplex numbers. Namely, while bicomplex numbers form a very nice commutative ring, they do not form a field, because not every non zero bicomplex number has a multiplicative inverse, and therefore we introduce the study of the zero-divisors in \(\mathbb{B }\mathbb{C }.\) It turns out that these zero-divisors play a central role, which is at first just intuited by looking at the so-called idempotent representation of bicomplex numbers. This section concludes with a study of the various multiplicative structures that can be induced on \(\mathbb{B }\mathbb{C }\); specifically we will consider how \(\mathbb{B }\mathbb{C }\) can be considered as an \(\mathbb{R }\)-linear space, a \(\mathbb{C }(\mathbf{i})\)-linear space, a \(\mathbb{C }(\mathbf{j})\)-linear space or finally as a \(\mathbb{D }\)-module.

The core of the paper, however, begins with Sect. 3, where we study the notion of derivability and holomorphy for bicomplex valued functions of a bicomplex variable. For simplicity we limit ourselves, in this section, to the considerations that can be obtained by limiting ourselves to the various cartesian representations of bicomplex numbers (i.e., we do not make reference, in this section, to the idempotent representation of bicomplex numbers). A fundamental remark, in this section, is the observation that the notion of derivability in the complex and real cases, are formulated in ways that are completely parallel to the bicomplex case. However, some of the most natural consequences that we derive in the real or complex case cannot be established in the bicomplex case (at least not in a completely symmetric way). This is clarified in Corollary 3.5, and the subsequent Remark 3.6. It is finally in this Sect. 3 that we establish various equivalent conditions to the notion of derivability (conditions that lead to the introduction of suitable Cauchy–Riemann systems). Note that in the case of quaternionic and real Clifford analysis, the theory of derivability of the corresponding functions is rather different from our situation (see, for instance [7, 15]).

In Sect. 4 we examine in detail the interplay between real differentiability and the derivability of bicomplex functions. Specifically, in the first subsection of Sect. 4, we express the usual notion of differentiability for a bicomplex function (thought of as a function of four real variables) in terms of its complex or hyperbolic differentiability; as we have seen, a bicomplex function can be thought of as a function of two complex variables (in \(\mathbb{C }(\mathbf{i})\) or in \(\mathbb{C }(\mathbf{j})\)) as well as a function of two hyperbolic variables, and therefore it is an interesting question to see how real differentiability relates to differentiability in terms of these other variables. As a consequence, we are able to express the real differentiability of a bicomplex function in four different languages: real, \(\mathbb{C }(\mathbf{i})\), \(\mathbb{C }(\mathbf{j})\), and hyperbolic.

But one may also want to look directly at bicomplex differentiability (i.e., differentiability when the function is regarded strictly as a function of one bicomplex variable). The result of this analysis gives the important equivalence, at least for \(\mathcal C ^1\) functions, between being \(\mathbb{B }\mathbb{C }\)-derivable and being a solution of a system of three Cauchy–Riemann like systems of differential equations.

The far reaching consequences of this result are explored in more depth in Sect. 5. All of this is of significant independent interest, but it also acts as the prologue for Sect. 6, where we study many variants of the Laplacian operator, in terms of the different variables which we can consider for a bicomplex function.

Finally, in Sect. 7, we go back to the issue of the meaning of bicomplex derivability and of bicomplex holomorphy. This time, however, we utilize the idempotent representation. As we had anticipated earlier in the paper, such a representation is quite powerful, and allows a deeper understanding of the notion of \(\mathbb{B }\mathbb{C }\)-holomorphy.

This paper should be seen as a complement to the theoretical description given by Price in [10], and we hope it will stimulate further researches in an area that appears still considerably understudied. In particular, it is important to note how our results clarify the relationship between \(\mathbb{B }\mathbb{C }\)-holomorphic functions and a suitable subset of holomorphic maps from \(\mathbb{C }^2\) to \(\mathbb{C }^2\), a topic which should be of considerable interest to specialists in several complex variables.

## 2 Bicomplex Numbers and Functions

*bicomplex*numbers is defined by

Within the set \(\mathbb{B }\mathbb{C }\) there are several subsets which one could legitimately recognize as isomorphic to the field of complex numbers: one of them is the set of those bicomplex numbers with \(z_2 =0\), namely those bicomplex numbers \(Z\) of the form \( Z=z_1 + \mathbf{j}0 = z_1\); this particular subset will be denoted, for obvious reasons, with the symbol \( \mathbb{C }( \mathbf{i})\). Since \(\mathbf{j}\) is another imaginary unit, we can define the set \( \mathbb{C }(\mathbf{j})\) within \(\mathbb{B }\mathbb{C }\) as \(\mathbb{C }(\mathbf{j}) := \{ z_1 + \mathbf{j}z_2 \mid z_1 , z_2 \in \mathbb{R }\, \}\). Of course, \( \mathbb{C }(\mathbf{i})\) and \( \mathbb{C }(\mathbf{j})\) are isomorphic fields but their immersion in \(\mathbb{B }\mathbb{C }\) creates an asymmetry, that will be of great importance in what follows.

*hyperbolic*numbers is defined intrinsically (independent of \(\mathbb{B }\mathbb{C }\)), as the set

*duplex*,

*double*or

*bireal*numbers. The addition and multiplication between hyperbolic numbers are defined in the obvious fashion, by replacing \(\mathbf{k}^2\) by 1 whenever it occurs.

Within \(\mathbb{B }\mathbb{C }\) a hyperbolic unit \(\mathbf{k}\) arises from the multiplication of the imaginary units \(\mathbf{i}\) and \(\mathbf{j}\), i.e., \(\mathbf{k}=\mathbf{i}\mathbf{j}\). Thus, we find that there is a subset in \(\mathbb{B }\mathbb{C }\) which is isomorphic as a ring to the set of hyperbolic numbers \(\mathbb{D }\), and which inherits all the algebraic definitions, operations and properties from \(\mathbb{B }\mathbb{C }\).

*bar*-conjugation is with respect to the imaginary unit \(\mathbf{i}\), the \(\dagger \)-conjugation is with respect to the imaginary unit \(\mathbf{j}\), and the \(*\)-conjugation is with respect to both \(\mathbf{i}\) and \(\mathbf{j}\).

But within \(\mathbb{B }\mathbb{C }\) there is also a hyperbolic unit. Since the (hyperbolic) *conjugate* of a hyperbolic number \(\mathfrak z =x+\mathbf{k}y\) is defined by \(\mathfrak z ^\diamond :=x - \mathbf{k}y\), we obtain that for a bicomplex number of the form \(\mathfrak z = x + \mathbf{i}\mathbf{j}y\) we have \({\overline{\mathfrak{z }}} = \mathfrak z ^\dagger = x - \mathbf{i}\mathbf{j}y = \mathfrak z ^\diamond \).

Note that working with just the complex numbers in \(\mathbb{C }(\mathbf{j})\), we will use the \(*\)-conjugation as the intrinsic conjugation on \(\mathbb{C }(\mathbf{j})\).

*hyperbolic modulus*as \(|\mathfrak z |_\mathbb{D }^2:=x^2-y^2\). If we now regard the hyperbolic number \(\mathfrak z \) as a bicomplex number, i.e., we write \(\mathfrak z =z_1+\mathbf{j}z_2=( x_1 + 0\mathbf{i}) + \mathbf{j}( 0 + \mathbf{i}y_2 ) \in \mathbb{B }\mathbb{C }\), one obtains the following relations among its various bicomplex moduli:

*invertible*if and only if \( Z\cdot Z^\dagger =z_1^2+z_2^2\ne 0\), and the

*inverse*of an invertible bicomplex number \(Z\) is given by the bicomplex number

*zero-divisor*, because \(Z\cdot Z^\dagger =0\). It is easy to show that all zero-divisors in \(\mathbb{B }\mathbb{C }\) are of the form: \(Z=\lambda (1\pm \mathbf{i}\mathbf{j}),\) where \(\lambda \) runs the whole set \( \mathbb{C }( \mathbf{i}) {\setminus } \{0\}\). An equivalent description of zero-divisors can also be obtained using the other writings of \(Z\). We denote the set of all zero-divisors in \(\mathbb{B }\mathbb{C }\) by \(\mathfrak S \), and we set \(\mathfrak S _0:=\mathfrak S \cup \{0\}\).

*idempotent representation*of the bicomplex number \(Z\). The importance of this representation lies in the fact that addition and multiplication of bicomplex numbers can be realized term-by-term in the idempotent representation. Analogously, the Euclidean norm of \(Z\) in terms of the idempotent components can be computed term-by-term as

We conclude this section with a short mention of the various algebraic structure that multiplication induces on \(\mathbb{B }\mathbb{C }\), depending on how we consider it.

In particular, the fact that the rings \(\mathbb{R }\), \(\mathbb{C }(\mathbf{i})\), \(\mathbb{C }(\mathbf{j})\), \(\mathbb{D }\) are subrings of the ring \(\mathbb{B }\mathbb{C }\), implies that \(\mathbb{B }\mathbb{C }\) can be seen as a module over each one of these subrings, and of course, it is a module over itself. Since \(\mathbb{R }\), \(\mathbb{C }(\mathbf{i})\) and \(\mathbb{C }(\mathbf{j})\) are fields, \(\mathbb{B }\mathbb{C }\) is a real linear space, a \(\mathbb{C }(\mathbf{i})\) complex linear space and a \(\mathbb{C }(\mathbf{j})\) complex linear space.

Formula (2.8) yields an isomorphism of real spaces between \(\mathbb{B }\mathbb{C }\) and \(\mathbb{R }^4\), which maps the bicomplex numbers \(1, \, \mathbf{i}, \, \mathbf{j}, \, \mathbf{k}\) into the canonical basis of \(\mathbb{R }^4\). Similarly, equation (2.2) yields an isomorphism between \(\mathbb{B }\mathbb{C }\) as a \( \mathbb{C }(\mathbf{i})\) linear space and \( \mathbb{C }^2(\mathbf{i})\), where the bicomplex numbers \(1, \, \mathbf{j}\) are mapped into the canonical basis of \(\mathbb{C }^2 (\mathbf{i})\). Seeing now \(\mathbb{B }\mathbb{C }\) as a \(\mathbb{C }(\mathbf{j})\)-linear space and using (2.3), we obtain an isomorphism which sends the bicomplex numbers \(1\) and \(\mathbf{i}\) into the canonical basis in \(\mathbb{C }^2(\mathbf{j})\).

Obviously these last two isomorphisms are different, which clarifies once again the fact that inside \(\mathbb{B }\mathbb{C }\), the “complex sets” \( \mathbb{C }^2(\mathbf{i}) \) and \(\mathbb{C }^2(\mathbf{j}) \) play different roles. A simple example that underscores this situation is the fact that the set \(\{1 , \, \mathbf{i}\}\) is linearly independent if we consider \(\mathbb{B }\mathbb{C }\) as a \( \mathbb{C }(\, \mathbf{j}) \)-linear space, but the same set is linearly dependent in the \( \mathbb{C }( \mathbf{i}) \)-linear space \(\mathbb{B }\mathbb{C }\).

Consider now the set \( \mathbb{D }^2 = \mathbb{D }\times \mathbb{D }\). With the component-wise addition inherited from \(\mathbb{D }\), one sees that \(\mathbb{D }^2\) is an additive abelian group. Defining also the component-wise multiplication by the scalars from \(\mathbb{D }\), \(\, \mathbb{D }^2\) becomes a hyperbolic module. Taking into account that \(\mathbb{B }\mathbb{C }\) is a module over \(\mathbb{D }\) and that it is also a ring, we will say that \(\mathbb BC \) is a commutative algebra over \(\mathbb D \), or a \(\mathbb D \)-algebra. Finally note that the set \(\mathbb BC \) is a bicomplex algebra. Finally note that the set \(\mathbb{B }\mathbb{C }\) is a bicomplex algebra.

## 3 Bicomplex Derivability and Bicomplex Holomorphy: The Case of Cartesian Representations

**Definition 3.1**

*Example 3.2*

*Remark 3.3*

We continue now with the definition of the derivative of a bicomplex function \(F:\Omega \subset \mathbb{B }\mathbb{C }\rightarrow \mathbb{B }\mathbb{C }\) of one bicomplex variable \(Z\) as follows (see, for instance [10], and references therein):

**Definition 3.4**

*derivative*\(F^{\prime }(Z_0)\) of the function \(F\) at a point \(Z_0\in \Omega \) is the limit, if it exists,

*derivable*at \(Z_0\).

**Corollary 3.5**

*Remark 3.6*

It is necessary to make a comment here. Traditionally, see e.g. [18, p. 138 and p. 432, if \(h\) is either a real or a complex increment, the symbol \(\mathfrak o (h)\) is used to indicate any expression of the form \(\alpha (h)|h|\) for \(\lim _{h \rightarrow 0} \alpha (h)=0.\) Since, both in the real and in the complex case, the expression \(\frac{|h|}{h}\) remains bounded when \(h \rightarrow 0,\) it is clear that one could replace \(\alpha (h)|h|\) by \(\alpha (h)h\) in the expression of \(\mathfrak o \). However, the situation is quite different in the bicomplex case. Here, the expression \(\frac{|H|}{H}\) is not bounded when \(H \rightarrow 0,\) and therefore we need to carefully distinguish the two expressions. In accordance with the usual notation, we will always use \(\mathfrak o (H)\) to denote a function of the form \(\alpha (H)|H|\), and therefore the expression in the previous corollary is not, in general, \(\mathfrak o (H).\) This distinction is at the basis of the notions of weak and strong Stoltz conditions for bicomplex functions, which are introduced by Price in [10]. We will discuss in detail how those conditions compare with our analysis elsewhere.

*Remark 3.7*

The existence of the derivative \(F^{\prime }(Z_0)\) has important consequences, as we show below.

**Theorem 3.8**

- 1.
The real partial derivatives \(\frac{\partial F}{\partial x_\ell }(Z_0)\) and \(\frac{\partial F}{\partial y_\ell }(Z_0)\) exist, for \(\ell =1,2\).

- 2.The real partial derivatives satisfy the following identities:$$\begin{aligned} F^{\prime }(Z_0)=\frac{\partial F}{\partial x_1}(Z_0) = -\mathbf{i}\frac{\partial F}{\partial y_1}(Z_0) = -\mathbf{j}\frac{\partial F}{\partial x_2}(Z_0) = \mathbf{k}\frac{\partial F}{\partial y_2}(Z_0). \end{aligned}$$(3.11)

*Proof*

The rest of the proof follows by considering the three specific forms of the increment \(H\) along the units \(\mathbf{i}, \mathbf{j}, \mathbf{k}\), i.e., \(H = \mathbf{i}y_1\), with \(y_1\ne 0\); \(H = \mathbf{j}x_2\), with \(x_2\ne 0\); \(H = \mathbf{k}y_2\), with \(y_2\ne 0\), and noting that all are invertible bicomplex numbers.\({\square }\)

Let us write the bicomplex function as \(F=f_{11}+\mathbf{i}f_{12} +\mathbf{j}f_{21} +\mathbf{k}f_{22}\) in terms of its real components, which are all real functions of a bicomplex variable. An immediate consequence of the theorem above is:

**Corollary 3.9**

*Proof*

The proof relies on a direct computation using the equalities (3.11) written in terms of \(f_{k\ell }\).\({\square }\)

*Remark 3.10*

The reader should notice that the special form of the real Jacobi matrix above encodes several Cauchy–Riemann type conditions on (certain pairs of) the real functions \(f_{k\ell }\), a fact which we will exploit in detail below.

*Remark 3.11*

As a next step, we investigate the consequence of the existence of \(F^{\prime }(Z_0)\) in terms of the complex variables \(z_1,z_2\in \mathbb{C }(\mathbf{i})\), where we write the bicomplex variable as \(Z=z_1+\mathbf{j}z_2\). We prove the following

**Theorem 3.12**

- 1.
The \(\mathbb{C }(\mathbf{i})\)-complex partial derivatives \(F^{\prime }_{z_\ell }(Z_0)\) exist, for \(\ell =1,2\).

- 2.The complex partial derivatives above verify the identity:which is equivalent to the \(\mathbb{C }(\mathbf{i})\)-$$\begin{aligned} F^{\prime }(Z_0) = F^{\prime }_{z_1}(Z_0) = -\mathbf{j}F^{\prime }_{z_2}(Z_0), \end{aligned}$$(3.14)
*complex Cauchy–Riemann system*for \(F\) (at \(Z_0\)), also called the*generalized Cauchy–Riemann system*in [13]:$$\begin{aligned} f^{\prime }_{1,z_1}(Z_0) = f^{\prime }_{2,z_2}(Z_0), \qquad f^{\prime }_{1,z_2}(Z_0) = -f^{\prime }_{2,z_1}(Z_0) . \end{aligned}$$(3.15)

*Proof*

*complex*Cauchy–Riemann conditions for \(F\) at \(Z_0\).\({\square }\)

**Corollary 3.13**

*Proof*

Let us now express the bicomplex variable as \(Z=\zeta _1+\mathbf{i}\,\zeta _2\), and the bicomplex function as \(F=\rho _1+\mathbf{i}\rho _2\), where \(\rho _1=f_{11}+\mathbf{j}f_{21}\) and \(\rho _2=f_{12}+\mathbf{j}f_{22}\) are \(\mathbb{C }(\mathbf{j})\)-valued functions. We prove the following

**Theorem 3.14**

- 1.
The \(\mathbb{C }(\mathbf{j})\)-complex partial derivatives \(F^{\prime }_{\zeta _\ell }(Z_0)\) exist, for \(\ell =1,2\).

- 2.The complex partial derivatives above verify the equality:which is equivalent to the \(\mathbb{C }(\mathbf{j})\)-$$\begin{aligned} F^{\prime }(Z_0) = F^{\prime }_{\zeta _1}(Z_0) = -\mathbf{i}F^{\prime }_{\zeta _2}(Z_0), \end{aligned}$$(3.22)
*complex Cauchy–Riemann system*(at \(Z_0\)):$$\begin{aligned} \rho ^{\prime }_{1,\zeta _1}(Z_0) = \rho ^{\prime }_{2,\zeta _2}(Z_0), \qquad \rho ^{\prime }_{1,\zeta _2}(Z_0) = -\rho ^{\prime }_{2,\zeta _1}(Z_0) . \end{aligned}$$(3.23)

*Proof*

**Corollary 3.15**

*Proof*

*Remark 3.16*

We express now the bicomplex variable \(Z=\mathfrak z _1+\mathbf{i}\mathfrak z _2\), where \(\mathfrak z _1 = x_1+\mathbf{k}y_2\) and \(\mathfrak z _2= y_1+\mathbf{k}(-x_2)\) are hyperbolic numbers, and the function \(F=\mathfrak f _1+\mathbf{i}\mathfrak f _2\), where \(\mathfrak f _1=f_{11}+\mathbf{k}f_{22}\) and \(\mathfrak f _2=f_{12}+\mathbf{k}(-f_{21})\). We prove the following

**Theorem 3.17**

- 1.
The hyperbolic partial derivatives \(F^{\prime }_\mathfrak{z _\ell }\) exist, for \(\ell =1,2\).

- 2.The partial derivatives above verify the equalitywhich is equivalent to the following Cauchy–Riemann type system for the hyperbolic components of a bicomplex derivable function:$$\begin{aligned} F^{\prime }(Z_0) = F^{\prime }_\mathfrak{z _1}(Z_0) = -\mathbf{i}F^{\prime }_\mathfrak{z _2}(Z_0), \end{aligned}$$$$\begin{aligned} \mathfrak f ^{\prime }_{1,\mathfrak z _1}(Z_0) = \mathfrak f ^{\prime }_{2,\mathfrak z _2}(Z_0), \qquad \mathfrak f ^{\prime }_{1,\mathfrak z _2}(Z_0) = -\mathfrak f ^{\prime }_{2,\mathfrak z _1}(Z_0) . \end{aligned}$$(3.28)

*Proof*

**Corollary 3.18**

*Cauchy–Riemann type systems*with respect to both variables \(\mathfrak z _1,\mathfrak z _1\in \mathbb{D }\); in hyperbolic terms this is equivalent to:

*Proof*

*Remark 3.19*

Appealing again to formulas (2.5)–(2.7) which deal with the complex variables \(w_1, w_2, \omega _1, \omega _2\) and the hyperbolic variables \(\mathfrak w _1, \mathfrak w _2\), one may wonder: what about the Cauchy–Riemann conditions with respect to the corresponding partial derivatives? Remark explains how they can be obtained directly from the previous statements. We omit the details.

**Definition 3.20**

Let \(F\) be a bicomplex function defined on a non-empty open set \(\Omega \subset \mathbb{B }\mathbb{C }.\) If \(F\) has bicomplex derivative at each point of \(\Omega \), we will say that \(F\) is a *bicomplex holomorphic*, or \(\mathbb{B }\mathbb{C }\)-holomorphic, function.

Thus for a \(\mathbb{B }\mathbb{C }\)-holomorphic function \(F\) all the conclusions made in this section hold in the whole domain. Theorem 3.12 says that \(F\) is holomorphic with respect to \(z_1\) for any \(z_2\) fixed and \(F\) is holomorphic with respect to \(z_2\) for any \(z_1\) fixed. Thus, see for instance [6, pages 4-5], \(F\) is holomorphic in the classical sense of two complex variables. This implies immediately many quite useful properties of \(F\), in particular, it is of class \(\mathcal C ^\infty (\Omega )\), and the reader may compare this with Remark 3.7 where we were able, working with just one point, not with a domain, to state a weakened continuity at the point.

## 4 Interplay Between Real Differentiability and Derivability of Bicomplex Functions

### 4.1 Real Differentiability in Complex and Hyperbolic Terms

Note that in Sect. 3 we introduced the symbols \(F^{\prime }_{z_1}(Z)\) and \(F^{\prime }_{z_2}(Z)\) instead of the symbols \(\frac{\partial F}{\partial z_1}(Z)\) and \(\frac{\partial F}{\partial z_2}(Z)\) because the former are complex partial derivatives, defined, as usual, as limits of suitable difference quotients, meanwhile the latter indicates well known operators acting on \(\mathcal C ^1\)-functions. The relationship between these two notions is clarified by the following definition and theorem.

**Definition 4.1**

**Theorem 4.2**

A \(\mathcal C ^1\) bicomplex function \(F\) is \(\mathbb{C }(\mathbf{i})\) complex differentiable if and only if both its components \(f_1, f_2\) are holomorphic functions in the sense of two complex variables.

*Proof*

Similarly, the partial derivative \(F^{\prime }_{z_2}(Z)\) exists in \(\Omega \) if and only if the operator \(\frac{\partial }{\partial {\overline{z}}_2}\) annihilates the function \(F\); this is because we can take now \(h_1=0, h_2\ne 0\) in (4.2). Therefore \(F^{\prime }_{z_2}(Z)=\frac{\partial F}{\partial z_2}(Z)\).

Note that for an arbitrary \(\mathbb{C }(\mathbf{i})\) complex differentiable bicomplex function, in general, there is no relation between its complex partial derivatives.

### 4.2 Real Differentiability in Bicomplex Terms

Formula (4.1) as well as formulas (4.2), (4.4) and (4.5) express the real differentiability of a bicomplex function, although written in different languages: the first of them is in real language, the next two in complex (\(\mathbb{C }(\mathbf{i})\) and \(\mathbb{C }(\mathbf{j})\)) language and the last is given in the hyperbolic one. Now we are going to see what the bicomplex language will give.

As a consequence of the previous discussion, we obtain the following result:

**Theorem 4.3**

*Proof*

Since \(F\) is \(\mathbb{B }\mathbb{C }\) holomorphic, formula (3.9) holds for all \(H\notin \mathfrak S _0.\) But \(F\) is a \(\mathcal C ^1\) function, hence (4.9) holds as well for any \(H\ne 0,\) thus both formulas hold for non zero-divisors. Then (4.10) follows directly by recalling that both (3.9) and (4.9) are unique representations for a given function \(F\), and by comparing them.\({\square }\)

*Remark 4.4*

As we will show in Sect. 7, the converse of this result is true as well, but we need some additional steps before we can prove it.

In order to have more consistency with the previous reasonings of this section and in analogy with the cases of functions of real or complex variables, we introduce the following definition.

**Definition 4.5**

*bicomplex*(\(\mathbb{B }\mathbb{C }\)-)

*differentiable*in \(\Omega \) if

Note that in this definition \(H\) is allowed to be a zero-divisor but taking \(H\) in (4.11) to be any non zero-divisor, we see from (3.9) that \(\mathbb{B }\mathbb{C }\) differentiability implies \(\mathbb{B }\mathbb{C }\) derivability. The reciprocal statement is more delicate and will not be treated immediately.

It turns out that Theorem 4.3 has many deep and far reaching consequences which we will discuss in the next section.

## 5 Various Interpretations and Consequences of Theorem 4.3

**Proposition 5.1**

A function \(F= f_1 +\mathbf{j}f_2 :\Omega \subset \mathbb{B }\mathbb{C }\rightarrow \mathbb{B }\mathbb{C }\) of class \(\mathcal{C }^1\) is \(\mathbb{B }\mathbb{C }\)-holomorphic if and only if, seen as a mapping from \(\Omega \subset \mathbb{C }^2(\mathbf{i})\rightarrow \mathbb{C }^2(\mathbf{i})\), it is a holomorphic mapping with its components related by the Cauchy–Riemann type conditions (5.1).

In other words, the theory of bicomplex holomorphic functions can be seen as a theory of a proper subset of holomorphic mappings in two complex variables. Each equation in Theorem 4.3 plays a different role: two of them together guarantee the holomorphy of the \(\mathbb{C }(\mathbf{i})\) complex components and the third one provides the relation between them.

It is remarkable that the operator \(\frac{\partial }{\partial Z^\dagger }\) arises in the works of Ryan [13] about complex Clifford analysis as a Cauchy–Riemann operator which is defined directly on holomorphic mappings with values in a complex Clifford algebra.

**Proposition 5.2**

A function \(F=g_1 +\mathbf{i}g_2:\Omega \subset \mathbb{B }\mathbb{C }\rightarrow \mathbb{B }\mathbb{C }\) of class \(\mathcal{C }^1\) is \(\mathbb{B }\mathbb{C }\)-holomorphic if and only if, seen as a mapping from \(\Omega \subset \mathbb{C }^2(\mathbf{j})\rightarrow \mathbb{C }^2(\mathbf{j})\), it is a holomorphic mapping with its components related by the Cauchy–Riemann type conditions (5.2).

**Proposition 5.3**

A function \(F=\mathfrak u _1 + \mathbf{i}\mathfrak u _2:\Omega \subset \mathbb{B }\mathbb{C }\rightarrow \mathbb{B }\mathbb{C }\) of class \(\mathcal{C }^1\) is \(\mathbb{B }\mathbb{C }\)-holomorphic if and only if, seen as a mapping from \(\Omega \subset \mathbb{D }^2\rightarrow \mathbb{D }^2\), it is a hyperbolic holomorphic mapping with its components related by the Cauchy–Riemann type conditions (5.3).

## 6 Second Order Complex and Hyperbolic Differential Operators

We have shown above that the theory of bicomplex holomorphic functions has deep similarities with its one complex variable counterpart, so let us look for analogues of the harmonic functions for it, or equivalently, for analogues of the real Laplacian.

The direct relations between the null solutions of any of the operators \(\Delta _{\mathbb{C }^2(\mathbf{j})}\) and \(\Delta _{\mathbb{D }^2}\) and the \(\mathbb{B }\mathbb{C }\)-holomorphic function theory is established in the same way as we did this for the operator \(\Delta _{\mathbb{C }^2(\mathbf{i})}\).

## 7 Bicomplex Holomorphy and Bicomplex Derivability: The Case of Idempotent Representation

*Remark 7.1*

Note that the above calculations show that, as it is well known, the bicomplex function \(F\) of class \(\mathcal C ^1\), seen as a mapping from \(\mathbb{C }^2(\mathbf{i})\rightarrow \mathbb{C }^2(\mathbf{i})\) is holomorphic with respect to \(\beta _q\) (\(q = 1,2\)) if and only if \(\frac{\partial F}{\partial {\overline{\beta }}_q}(Z)=0\) in \(\Omega \). Note that if \(F\) is a bicomplex function, and we express it in cartesian coordinates, it turns out that \(F\) is \(\mathbb{B }\mathbb{C }\)-holomorphic if and only if its components are holomorphic as functions of two complex variables and satisfy a Cauchy–Riemann type relation between them. As we will show later, this is definitely not the case when we express \(F\) in the idempotent representation. In this case, \(\mathbb{B }\mathbb{C }\)-holomorphy will be equivalent to the requirement that each component is a holomorphic function of a single complex variable and there are no relations between the components.

**Theorem 7.2**

The \(\mathcal C ^1\) function \(F\) is \(\mathbb{B }\mathbb{C }\)-holomorphic if and only if the three coefficients of \(H^\dagger , {\overline{H}}\) and \(H^*\) are all zero for any \(Z\) in \(\Omega \).

*Proof*

*if*direction follows as in Theorem 4.3. Specifically, since \(F\) is \(\mathbb{B }\mathbb{C }\)-holomorphic, formula (3.9) holds for all \(H\notin \mathfrak S _0.\) But \(F\) is a \(\mathcal C ^1\) function, hence (7.3) holds as well for any \(H\ne 0,\) thus both formulas hold for non zero-divisors. Then the result follows directly by recalling that both (3.9) and (7.3) are unique representations for a given function \(F\), and by comparing them. In order to prove the

*only if*, it is helpful to write explicitly the meaning of the vanishing of these coefficients, namely:

Now, by the properties of \(G_1\) and \(G_2\) we deduce that the right hand side has, for \( \mathfrak S \not \ni H \rightarrow 0\), the limit \(G^{\prime }_1(\beta _1)\mathbf{e}+ G^{\prime }_2(\beta _2)\mathbf{e^\dagger },\) which concludes the proof: the limit in the left hand-side exists also for any \(Z\in \Omega \) with \(\mathfrak S H\rightarrow 0\) and it coincides with the derivative \(F^{\prime }(Z)\) making \(F\)\(\mathbb{B }\mathbb{C }\)-holomorphic in \(\Omega \).\({\square }\)

As a matter of fact, the proof allows to make a more precise characterization of \(\mathcal C ^1\) functions which are \(\mathbb{B }\mathbb{C }\)-holomorphic.

**Theorem 7.3**

- (I)
The component \(G_1\), seen as a \(\mathbb{C }(\mathbf{i})\) valued function of two complex variables \((\beta _1,\beta _2)\) is holomorphic; what is more, it does not depend on the variable \(\beta _2\) and thus \(G_1\) is a holomorphic function of the variable \(\beta _1\).

- (II)
The component \(G_2\), seen as a \(\mathbb{C }(\mathbf{i})\) valued function of two complex variables \((\beta _1,\beta _2)\) is holomorphic; what is more, it does not depend on the variable \(\beta _1\) and thus \(G_2\) is a holomorphic function of the variable \(\beta _2\).

*Remark 7.4*

The functions \(G_1\) and \(G_2\) are independent in the sense that there are no Cauchy–Riemann type conditions relating them.

We are in a position now to prove that the converse to Theorem 4.3 is true as well.

**Theorem 7.5**

Given \(F \in \mathcal C ^1 ( \Omega , \, \mathbb{B }\mathbb{C }) \), then condition (4.10) implies that \(F\) is \(\mathbb{B }\mathbb{C }\)-holomorphic.

*Proof*

If (4.10) holds then a direct computation shows that all the three formulas in (7.4) are true, and by Theorem 7.2 \(F\) is \(\mathbb{B }\mathbb{C }\)-holomorphic.\({\square }\)

**Corollary 7.6**

*Remark 7.7*

Although formula (7.3) is quite similar to formula (4.9) its consequences for the function \(F\) are paradoxically different: meanwhile formula (4.9) has allowed to conclude that the cartesian components \(f_1, f_2\) are holomorphic functions of two complex variables which are not independent, formula (7.3) explains us that the idempotent components \(G_1, G_2\) are usual holomorphic functions of one complex variable which are, besides, independent.

*Remark 7.8*

*Remark 7.9*

The same analysis can be done for the idempotent representation with \(\mathbb{C }(\mathbf{j})\) coefficients.

*Remark 7.10*

Recall that in Sect. 3 we departed from the cartesian representation of bicomplex numbers and we investigated many properties of derivable bicomplex functions, in particular such functions proved to have complex partial derivatives with respect to \(z_1, z_2\). This approach fails immediately when one tries to apply it to the case of idempotent representation: the matter is that the definition of the derivative excludes precisely the values of \(H\) which are necessary for the complex partial derivatives with respect to \(\beta _1, \beta _2\). But in the proof of Theorem 4.3 we have proved, as a matter of fact, that such partial derivatives of a \(\mathbb{B }\mathbb{C }\)-holomorphic functions do exist and, moreover, \( \frac{\partial F}{\partial \beta _1} (Z) = G_1^{\prime } (\beta _1) \cdot \mathbf{e}\) and \( \frac{\partial F}{\partial \beta _2} (Z) = G_2^{\prime } (\beta _2) \cdot \mathbf{e^\dagger }\).