Scator Holomorphic Functions

Abstract

Scators form a linear space equipped with a specific non-distributive product. In the elliptic case they can be interpreted as a kind of hypercomplex number. The requirement that the scator partial derivatives are direction-independent leads to a generalization of the Cauchy–Riemann equation and to scator holomorphic functions. In this paper we find a complete set of $C^2$-solutions to the generalized Cauchy–Riemann system in the $(1+n)$-dimensional elliptic scator space. For any $n⩾2$ this set consists of three classes: components exponential functions (already known), a new class of affine linear functions, and some exceptional solutions parameterized by arbitrary functions of one variable. We show, however, that the last class of solutions is not scator holomorphic and the generalized Cauchy–Riemann system should be supplemented with additional constraints to avoid such spurious solutions. The obtained family of scator holomorphic functions, although relatively narrow, is greater than that of analogous functions in quaternionic or Clifford analysis.

1. Introduction

Scators were introduced and discussed by Manuel Fernández-Guasti [1,2]. They form a linear space with a specific multiplicative structure. In the $(1+n)$-dimensional elliptic case, the scator product of two scators, $\stackrel{o}{a}:=(a_0;a_1,\dots ,a_n)$ and $\stackrel{o}{b}:=(b_0;b_1,\dots ,b_n)$, is given by $\stackrel{o}{u}:=(u_0;u_1,\dots ,u_n)$, where

$$\begin{array}{c}{\displaystyle u_0=a_0{b}_0\prod _{k=1}^n\left(1-\frac{a_k{b}_k}{a_0{b}_0}\right),}\hfill \\ {\displaystyle u_k=\frac{a_k{b}_0+b_k{a}_0}{a_0{b}_0-a_k{b}_k}{u}_0(k=1,\dots ,n),}\hfill \end{array}$$

(1)

provided that $a_0\ne 0$ and $b_0\ne 0$ (a more general case is presented and discussed in [1]). In the hyperbolic case all minuses should be replaced by pluses in the above formula. Mixed cases can be considered as well. The scator product is non-distributive, although a distributive approach has been recently proposed, see [3,4]. It consists in embedding scators in a larger space, where the multiplication is distributive. In addition, another example of a non-distributive algebra of hypercomplex numbers has been proposed recently [5].

In the hyperbolic case scators have a potential physical interpretation and applications related to some deformations of special relativity [6,7], while the elliptic case is a source of new (non-distributive) hypercomplex numbers and hyperholomorphic functions [2,8].

Any structure of hypercomplex numbers, usually expressed in terms of Clifford numbers (see, e.g., [9]), leads to a natural question of defining analogues of holomorphic functions [10,11,12,13]. In this paper, following [8] (see also [14]), by “holomorphy” we mean the existence of a direction-independent partial derivative at any point. Scator differentiability of this kind leads to a system of partial differential equations, which can be considered as a generalization of the Cauchy–Riemann equations of complex analysis (see Theorem 3.1 in [8]):

$$\begin{array}{c}{\displaystyle \frac{\partial u_0}{\partial x_0}=\frac{\partial u_j}{\partial x_j},\frac{\partial u_j}{\partial x_0}=-\frac{\partial u_0}{\partial x_j},}\hfill \\ {\displaystyle \frac{\partial u_0}{\partial x_j}\frac{\partial u_0}{\partial x_m}=-\frac{\partial u_j}{\partial x_j}\frac{\partial u_j}{\partial x_m}}\hfill \end{array}$$

(2)

for all $m\ne j$, where m and j take values from 1 to n. In Section 2 we show that in order to avoid spurious solutions an additional assumption must be added, namely Equation (5). The generalized Cauchy–Riemann equations (2) consist of linear equations (looking like n copies of the Cauchy–Riemann equations) and a set of nonlinear equations (for $n>1$). In the case $n=1$, the system (2) reduces to the standard Cauchy–Riemann equations because the nonlinear part of the system (2) is missing (both m and j can assume only one value, so they cannot be different). Therefore we obtain standard holomorphic functions on $\mathbb{C}$.

In this paper we are going to solve the open problem of finding all solutions of the system (2) in the case of arbitrary n. The case $n=2$ has already been solved recently [15], while for $n>2$ two classes of solutions have been reported earlier: a very narrow class of linear affine functions [8] and components exponential functions [16,17]. As a result we obtain a new large class of linear affine holomorphic functions. We also show that, similarly to the case $n=2$, there exists a third class of solutions to the system (2). However, this third class turns out to consist of spurious solutions, which are not scator holomorphic.

2. Implicitly Omitted Assumptions

In this paper we work within the theoretical framework outlined in [8]. However, some points have to be clarified. First of all, the scator differentiability is referred to as the direction-independence of the scator derivative, but in all proofs only the equality of all partial derivatives is considered. Such a concept of the scator differentiability is weaker than both Gâteaux and Fréchet derivatives. The same concerns the scator holomorphy (i.e., differentiability in some open domain).

Equating scator partial derivatives with respect to scalar and director components (see Equations (11) and (12) in [8]), we obtain

$$u{,}_0+\sum _{j=1}^n{u}_j{,}_0{\stackrel{o}{\mathit{e}}}_j=u_m{,}_m-u{,}_m{\stackrel{o}{\mathit{e}}}_m+{\displaystyle \frac{u_m{,}_m}{u{,}_m}}\sum _{j\ne m}{u}_j{,}_m{\stackrel{o}{\mathit{e}}}_j(m=1,2,\dots ,n),$$

(3)

where by ${\stackrel{o}{\mathit{e}}}_j$ we denote basis elements in the scator space. Here and in the sequel we use the following notation:

$$u:=u_0,u{,}_0:={\displaystyle \frac{\partial u_0}{\partial x_0}},u{,}_j:={\displaystyle \frac{\partial u_0}{\partial x_j}},u_j{,}_m:={\displaystyle \frac{\partial u_j}{\partial x_m}},$$

(4)

for $j,m=1,\dots ,n$. The Einstein summation convention is never used in this paper.

Fernández-Guasti assumed that functions $u_0,u_1,\dots u_n$ are of class $C^1$ and from the system (4) he derived (2). However, additional assumptions are apparent upon inspecting this proof, namely

$${\displaystyle \frac{\partial u_0}{\partial x^m}}\ne 0(\mathrm{for}m=1,2,\dots ,n),$$

(5)

in the considered domain. Theorem 3.1 in [8] states that scator-differentiable functions must satisfy the Equation (2), which is correct, although spurious solutions are still possible. By adding condition (5), we obtain a stronger statement: functions satisfying both the generalized Cauchy–Riemann system (2) and the constraints (5) are scator holomorphic.

3. Derivation of Solutions to the Generalized Cauchy–Riemann System

To obtain all solutions, it is necessary to take into account all cases, including exceptional and singular ones. We always assume $n⩾2$, because the case $n=1$ reduces to the well-known classical Cauchy–Riemann equations. We perform standard local computation in some connected open domain $\mathsf{\Omega }\subset R^{n+1}$. We assume that functions $u_0,u_1,\dots ,u_n$ are of class $C^2$

3.1. The Case  ${\displaystyle \frac{\partial u_0}{\partial x_0}\ne 0}$

In this case we can easily transform (2) into the following form:

$$\begin{array}{c}{\displaystyle \frac{\partial u_j}{\partial x_0}=-\frac{\partial u_0}{\partial x_j},\frac{\partial u_j}{\partial x_j}=\frac{\partial u_0}{\partial x_0},(j=1,\dots ,n),}\hfill \\ {\displaystyle \frac{\partial u_j}{\partial x_m}=-\frac{{\displaystyle \frac{\partial u_0}{\partial x_j}\frac{\partial u_0}{\partial u_m}}}{{\displaystyle \frac{\partial u_0}{\partial x_0}}},(j,m=1,\dots ,n),m\ne j,}\hfill \end{array}$$

(6)

where the right-hand sides depend only on $u_0$ and its derivatives. The compatibility conditions for the existence of a solution $u_1,\dots ,u_n$ (i.e., $u_k{,}_{\mu \nu }=u_k{,}_{\mu \nu }$ for $k=1,\dots n$ and $\mu ,\nu =0,1,\dots ,n$) are given by

$$\begin{array}{c}\left(u{,}_j\right){,}_j=-\left(u{,}_0\right){,}_0,(j=1,\dots ,n),\hfill \\ {\displaystyle \left(u{,}_j\right){,}_m=\left(\frac{u{,}_{j}u{,}_m}{u{,}_0}\right){,}_0,(j,m=1,\dots ,n),m\ne j,}\hfill \\ {\displaystyle \left(u{,}_0\right){,}_m=-\left(\frac{u{,}_{j}u{,}_m}{u{,}_0}\right){,}_j,(j,m=1,\dots ,n),m\ne j,}\hfill \end{array}$$

(7)

The compatibility conditions (7) can be easily reduced to the following form:

$$\begin{array}{c}u{,}_{jj}+u{,}_{00}=0(j=1,\dots ,n),\hfill \\ {\displaystyle u{,}_{jm}=\frac{u{,}_{j0}u{,}_{m}u{,}_0+u{,}_{m0}u{,}_{j}u{,}_0-u{,}_{00}u{,}_{m}u{,}_j}{{\left(u{,}_0\right)}^2},(m,j=1,\dots ,n),m\ne j,}\hfill \\ {\displaystyle u{,}_{0m}=\frac{u{,}_{0j}u{,}_{j}u{,}_m-u{,}_{jj}u{,}_{m}u{,}_0-u{,}_{mj}u{,}_{j}u{,}_0}{{\left(u{,}_0\right)}^2},(m,j=1,\dots ,n),m\ne j.}\hfill \end{array}$$

(8)

The system (8) consists of partial differential equations for only one field, namely $u_0\equiv u$. Therefore, we plan first to solve the system (8) and then (knowing u) to integrate the linear equations of the first order for the remaining fields $u_j$ ($j=1,\dots ,n$).

3.1.1. Solving the Compatibility Conditions (8)

Substituting $u{,}_{jj}$ and $u{,}_{jm}$ from the first two equations of (8) into the last Equation (8), we get

$$u{,}_{0m}={\displaystyle \frac{u{,}_{j0}u{,}_{j}u{,}_m+u{,}_{00}u{,}_{m}u{,}_0}{{\left(u{,}_0\right)}^2}}-{\displaystyle \frac{u{,}_j}{u{,}_0}}\left({\displaystyle \frac{u{,}_{j0}u{,}_{m}u{,}_0+u{,}_{m0}u{,}_{j}u{,}_0-u{,}_{00}u{,}_{m}u{,}_j}{{\left(u{,}_0\right)}^2}}\right),$$

(9)

for $m,j=1,\dots ,n$ ($m\ne j$). Canceling the first term with the third term in Equation (9), we obtain

$$u{,}_{0m}={\displaystyle \frac{u{,}_{00}u{,}_m}{u{,}_0}}-{\displaystyle \frac{{\left(u{,}_j\right)}^2}{u{,}_0}}\left({\displaystyle \frac{u{,}_{m0}u{,}_0-u{,}_{00}u{,}_m}{{\left(u{,}_0\right)}^2}}\right),$$

(10)

which is equivalent to

$$\left(1+{\left({\displaystyle \frac{u{,}_j}{u{,}_0}}\right)}^2\right)u{,}_{0m}={\displaystyle \frac{u{,}_{00}u{,}_m}{u{,}_0}}\left(1+{\left({\displaystyle \frac{u{,}_j}{u{,}_0}}\right)}^2\right).$$

(11)

Hence we have $u{,}_{0m}u{,}_0=u{,}_{00}u{,}_m$, which, given that $u{,}_0\ne 0$, can be written as

$$\left({\displaystyle \frac{u{,}_m}{u{,}_0}}\right){,}_0=0.$$

(12)

Thus, as a necessary consequence of the assumption $u{,}_0\ne 0$, we get the following useful equations:

$$u{,}_m=u{,}_0{F}_m(x_1,\dots ,x_n)(m=1,\dots ,n),$$

(13)

where $F_m$ are functions that do not depend on $x_0$.

Differentiating Equation (13) with respect to $x_0$ and $x_j$, we obtain

$$\begin{array}{c}u{,}_{m0}=u{,}_{00}{F}_m(m=1,\dots ,n),\hfill \\ u{,}_{mj}=u{,}_{00}{F}_j{F}_m+u{,}_0{F}_m{,}_j(j,m=1,\dots ,n),\hfill \end{array}$$

(14)

where the case $m=j$ is included. Note that, since u is $C^2$, the functions $F_m$ are necessarily $C^1$. Now, we can simplify the system (8) using Equations (13) and (14):

$$\begin{array}{c}u{,}_{00}{F}_j{F}_j+u{,}_0{F}_j{,}_j+u{,}_{00}=0,\hfill \\ u{,}_{00}{F}_j{F}_m+u{,}_0{F}_m{,}_j=u{,}_{00}{F}_j{F}_m+u{,}_{00}{F}_m{F}_j-u{,}_{00}{F}_m{F}_j,\hfill \\ u{,}_{00}{F}_m=u_{00}{F}_j{F}_j{F}_m-(u{,}_{00}{F}_j{F}_j+u{,}_0{F}_j{,}_j)F_m-(u{,}_{00}{F}_j{F}_m+u{,}_0{F}_m{,}_j)F_j.\hfill \end{array}$$

(15)

Upon obvious cancellation of several terms, we obtain

$$\begin{array}{c}u{,}_{00}(1+F_j^2)+u{,}_0{F}_j{,}_j=0(j=1,\dots ,n),\hfill \\ u{,}_0{F}_m{,}_j=0(m,j=1,\dots ,n),m\ne j,\hfill \\ u{,}_0{F}_j{F}_m{,}_j=0(m,j=1,\dots ,n),m\ne j.\hfill \end{array}$$

(16)

The last equation is redundant, resulting directly from the second one. Therefore, the generalized Cauchy–Riemann system (8) reduces to

$${\displaystyle \frac{F_j{,}_j}{1+F_j^2}}=-{\displaystyle \frac{u{,}_{00}}{u{,}_0}},F_j=F_j\left(x_j\right),u,j=u{,}_0{F}_j(j=1,\dots ,n),$$

(17)

where, at the end, we repeated Equation (13). Hence,

$${\displaystyle \frac{F_j{,}_j}{1+F_j^2}}={\displaystyle \frac{F_m{,}_m}{1+F_m^2}}(\mathrm{for}k,m=1.\dots ,n)$$

(18)

and, by virtue of $F_j=F_j\left(x_j\right)$, both sides of (18) must be constant.

$${\displaystyle \frac{F_j{,}_j}{1+F_j^2}}=-{\omega }_0,(log|u{,}_0\left|\right){,}_0={\omega }_0({\omega }_0=\mathrm{const}).$$

(19)

The above equations for $F_j$ and u can be easily integrated. If ${\omega }_0\ne 0$, then

$$\begin{array}{c}{F}_j=-tan({\omega }_0{x}_j+a_j)(a_j=\mathrm{const})j=1,\dots ,n,\hfill \\ {\displaystyle u=\frac{1}{{\omega }_0}{e}^{{\omega }_0{x}_0}g(x_1,\dots ,x_n)+h(x_1,\dots ,x_n)}\hfill \end{array}$$

(20)

where g and h are functions of n variables. The last equation (for $u{,}_j$) of (17) yields

$${\displaystyle \frac{1}{{\omega }_0}}{e}^{{\omega }_0{x}_0}g{,}_j+h{,}_j=-e^{{\omega }_0{x}_0}gtan({\omega }_0{x}_j+a_j).$$

(21)

Hence $h{,}_j=0$ (for $j=1,\dots ,n$), i.e.,

$$h(x_1,\dots ,x_n)=d_0,(d_0=\mathrm{const}),$$

(22)

and

$$g{,}_j=-{\omega }_{0}gtan({\omega }_0{x}_j+a_j)(j=1,\dots ,n),$$

(23)

which is equivalent to

$${\displaystyle \frac{\partial }{d{x}_j}}\left({\displaystyle \frac{g}{cos({\omega }_0{x}_j+a_j)}}\right)=0(j=1,\dots ,n).$$

(24)

Therefore

$$g=g_{0}cos({\omega }_0{x}_1+a_1)cos({\omega }_0{x}_2+a_2)\dots cos({\omega }_0{x}_n+a_n),$$

(25)

where $g_0$ is a constant, and, finally,

$$\begin{array}{c}{\displaystyle u=d_0+\frac{g_0}{{\omega }_0}{e}^{{\omega }_0{x}_0}\prod _{j=1}^{n}cos({\omega }_0{x}_j+a_j),}\hfill \\ u{,}_0=e^{{\omega }_0{x}_0}g.\hfill \end{array}$$

(26)

If ${\omega }_0=0$, then $F_j=c_j=\mathrm{const}$ and

$$u=x_{0}g(x_1,\dots ,x_n)+h(x_1,\dots ,x_n).$$

(27)

Similarly to the generic case, constraints on g and h follow from the last equation of (17):

$$x_{0}g{,}_j+h{,}_j=g{c}_j(j=1,\dots ,n).$$

(28)

Hence $g=g_0=\mathrm{const}$ and h is linear with respect to all variables:

$$u=d_0+g_0(x_0+c_1{x}_1+\dots +c_n{x}_n),$$

(29)

where $d_0$, $g_0$, and $c_j$ ($j=1,\dots ,n$) are constants.

3.1.2. Solving the Full System (6)

First, we consider the case $u{,}_0\ne 0$ (remember also that $u_0\equiv u$)

$$\begin{array}{c}{u}_j{,}_0=-u{,}_0{F}_j,\hfill \\ u_j{,}_j=u{,}_0,\hfill \\ u_j{,}_m=-u{,}_0{F}_j{F}_m,\hfill \end{array}$$

(30)

where $u{,}_0$ is given by (26). The first equation can be easily integrated,

$$u_j=-u{F}_j+D_j(x_1,\dots ,x_n)$$

(31)

Inserting (31) into the third equation of (30), we get $D_j{,}_m=0$ (for $m\ne j$); i.e.,

$$D_j=D_j\left(x_j\right).$$

(32)

Then the second equation of (30) reduces to

$$-u{,}_0{F}_j^2-u{F}_j{,}_j+D_j{,}_j=u{,}_0,$$

(33)

and, taking into account $u=d_0-{\left({\omega }_0\right)}^{-1}u{,}_0$ and $F_j{,}_j={\omega }_0(1+F_j^2)$ (see (19) and (26)), we get

$$d_0{F}_j{,}_j+D_j{,}_j=0.$$

(34)

Hence

$$D_j=d_0{F}_j+d_j(j=1,\dots ,n),$$

(35)

where $d_j$ are constants. Thus, by (31), we obtain

$$u_j=(d_0-u)F_j+d_j.$$

(36)

Therefore, in the case ${\omega }_0\ne 0$ we have

$$u_j=d_j+{\displaystyle \frac{g_0}{{\omega }_0}}{e}^{{\omega }_0{x}_0}tan({\omega }_0{x}_j+a_j)\prod _{k=1}^{n}cos({\omega }_0{x}_k+a_k).$$

(37)

For ${\omega }_0=0$ everything simplifies: $u{,}_0=g_0$, $F_j=c_j$, and the system (30) reduces to

$$\begin{array}{c}{u}_j{,}_0=-g_0{c}_j,\hfill \\ u_j{,}_j=g_0,\hfill \\ u_j{,}_m=-g_0{c}_j{c}_m,\hfill \end{array}$$

(38)

and its general solution is given by

$$u_j=d_j+g_0\left(-c_j{x}_0+x_j-c_j\sum _{{\displaystyle \genfrac{}{}{0.0pt}{}{k=1}{k\ne j}}}^n{c}_k{x}_k\right).$$

(39)

In other words, the scator holomorphic function given by (29) and (39) is just the affine map

$$u_{\mu }=d_{\mu }-g_0\sum _{\nu =0}^n{J}_{\mu \nu }{x}_{\nu }(\mu =0,1,\dots ,n),$$

(40)

where $J_{\mu \nu }$ are the entries of the following square matrix J of size ($n+1$):

$$J=\left(\begin{array}{cccccc}-1& -c_1& -c_2& -c_3& \dots & -c_n\\ c_1& -1& c_1{c}_2& c_1{c}_3& \dots & c_1{c}_n\\ c_2& c_2{c}_1& -1& c_2{c}_3& \dots & c_2{c}_n\\ c_3& c_3{c}_1& c_3{c}_2& -1& \dots & c_3{c}_n\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ c_n& c_n{c}_1& c_n{c}_2& c_n{c}_3& \dots & -1\end{array}\right),$$

(41)

which is proportional to the Jacobi matrix of the map (40). The matrix J is Lorentz symmetric; i.e., $GJ$ is symmetric for $G=\mathrm{diag}(-1,1,\dots ,1)$. Actually

$$J=G(\mathit{c}{\mathit{c}}^T-D)$$

(42)

where ${\mathit{c}}^T=(1,c_1,c_2,\dots ,c_n)$ and $D=I+\mathrm{diag}(-1,c_1^2,\dots ,c_n^2)$. Therefore, using the matrix determinant lemma or Cauchy’s formula for the determinant of a rank-one perturbation (see, e.g., Equation (0.8.5.11) in [18]), we obtain the formula for the determinant of J

$$detJ={(-1)}^{n+1}\prod _{k=1}^n(1+c_k^2),$$

(43)

and the characteristic equation for the eigenvalues of the matrix J in the compact form:

$$1+{\displaystyle \frac{1}{\lambda }}=\sum _{j=1}^n{\displaystyle \frac{c_j^2}{c_j^2+\lambda +1}}.$$

(44)

Note that in the case $n=1$, it reduces to the expected result: ${(1+\lambda )}^2+c_1^2=0$. What is more, $|detJ|>1$, which means that J is always invertible.

While the symmetric form of the Jacobi matrix J is intriguing, an interpretation and implications of this phenomenon in the context of the scator holomorphy are not yet clear.

3.2. The Case $u{,}_0=0$

In the case $u{,}_0=0$, we have to come back to the original system (2) taking into account that $u_j{,}_j=u{,}_0=0$. Unlike the previous subsection, we do not consider the compatibility conditions first, but immediately solve the full system (2), which in this case reduces to

$$\begin{array}{c}{u}_j{,}_j=0,\hfill \\ u_j{,}_0=-u{,}_j,\hfill \\ u{,}_{j}u{,}_m=0.\hfill \end{array}$$

(45)

The last equation of (45) implies that $u{,}_j$ is non-zero for at most one value of j, say $j=m$. In other words u depends only on one variable: $u={\rho }_m\left(x_m\right)$. Then the second equation yields

$$\begin{array}{c}{u}_m{,}_0=-{\rho }_m^{\prime }\left(x_m\right),\hfill \\ u_j{,}_0=0(j\ne m),\hfill \end{array}$$

(46)

which means that

$$\begin{array}{c}{u}_m=-x_0{\rho }_m^{\prime }\left(x_m\right)+{\psi }_m(x_1,\dots ,x_n),\hfill \\ u_j={\varphi }_{mj}(x_1,\dots ,x_n)(j\ne m),\hfill \end{array}$$

(47)

and, applying the first equation of (45), we get ${\rho }_m^{\prime \prime }=0$ and ${\psi }_m{,}_m=0$, which means that for any $m⩽n$ we have

$$\begin{array}{c}u=c_{1m}+c_{0m}{x}_m,\hfill \\ u_m=-c_{0m}{x}_0+{\varphi }_{mm}(x_1,\dots ,\widehat{x_m},\dots ,x_n),\hfill \\ u_j={\varphi }_{mj}(x_1,\dots ,\widehat{x_j},\dots ,x_n)(j\ne m),\hfill \end{array}$$

(48)

where $c_{0m}$ and $c_{1m}$ are constants, ${\psi }_m$ and ${\varphi }_{mj}$ are arbitrary functions of $n-1$ variables, and the notation $\widehat{x_m}$ means that ${\psi }_m$ does not depend on $x_m$ (and, in our case, can depend on all other variables). Two indices on $\varphi$ in the last line (${\varphi }_{mj}$) underline that for each m we have a solution $(u_0,u_1,\dots ,u_n)$, expressed by a different set of arbitrary functions. Finally, we introduced ${\varphi }_{mm}:={\psi }_m$.

We derived the full set of solutions in this case, but already from the last equation of (45) it follows that the assumption (5) is violated, so the solution (48) is spurious.

4. Final Results and Classification of Solutions

The results of previous section, expressed by the Formulas (26), (37), (40), and (48), can be summarized by the following theorem.

Theorem 1.

The full set of solutions to the generalized Cauchy–Riemann equations (2), supplemented by constraints (5), consists of two families.

• Components exponential functions:

  $$\begin{array}{c}{\displaystyle u_0=d_0+{\lambda }_0{e}^{{\omega }_0{x}_0}\prod _{j=1}^{n}cos({\omega }_0{x}_j+a_j),}\hfill \\ {\displaystyle u_k=d_k+{\lambda }_0{e}^{{\omega }_0{x}_0}tan({\omega }_0{x}_k+a_k)\prod _{j=1}^{n}cos({\omega }_0{x}_j+a_j),(k=1,\dots ,n),}\hfill \end{array}$$

  (49)

  where ${\lambda }_0$, $a_{\mu }$, and $d_{\mu }$ ($\mu =0,1,\dots ,n$) are real constants.
• Affine functions of the special form:

  $$u_{\mu }=d_{\mu }+{\lambda }_0\sum _{\nu =0}^n{J}_{\mu \nu }{x}_{\nu }(\mu =0,1,\dots ,n),$$

  (50)

  where ${\lambda }_0$ and $d_{\mu }$ are real constants and $J_{\mu \nu }$ are entries of the constant matrix J given by (41), which is expressed by constants $c_{\mu }$ ($\mu =0,1,\dots ,n$).

We point out that holomorphic affine linear functions found in [8] form only a narrow subclass of (50) corresponding to $c_{\mu }=0$ for $\mu =0,1,\dots ,n$.

The structure of the solution space of the generalized Cauchy–Riemann equations is essentially the same for all $n⩾2$. In particular, for $n=2$ the results of [15], although obtained in a slightly different way, coincide with the results presented above. It should be noted that paper [15] contains misprints in Formulas (7), (32), and (33). In these formulas, the functions $V_1$ and $V_2$ should depend only on z, whereas $W_1$ and $W_2$ should depend only on y. However, the solutions from the third class are spurious, see Section 3.2, and the indicated misprints appear only in formulas related to this class.

The system (2) is nonlinear for $n⩾2$, so linear combinations of solutions usually do not satisfy this system. However, one can easily show that homogeneous affine transformations (translations and homogeneous dilations) are symmetries of the system (2).

Theorem 2.

The generalized Cauchy–Riemann equations (2) are invariant with respect to homogeneous affine transformations both in dependent and independent variables:

$${\tilde{u}}_{\mu }=d_{\mu }+{\lambda }_0{u}_{\mu },{\tilde{x}}_{\mu }=a_{\mu }+{\omega }_0{x}_{\mu },$$

(51)

where ${\lambda }_0,{\omega }_0,a_{\mu },d_{\mu }\in \mathbb{R}$ (for $\mu =0,1,\dots ,n$), with ${\lambda }_0\ne 0$ and ${\omega }_0\ne 0$.

Many constants used in Theorem 1 (see Equations (49) and (50)) arise from these symmetries. We use similar notation in both theorems to emphasize this relationship. The classification of scator holomorphic functions modulo the affine symmetries reduces to the components exponential function (denoted by cexp, see [16]):

$$\mathrm{cexp}(\stackrel{o}{x})=\exp \left(x_0\right)\left(1+\sum _{k=1}^n{\stackrel{o}{\mathbf{e}}}_{k}tan\left(x_k\right)\right)\prod _{j=1}^{n}cos\left(x_j\right)$$

(52)

and the following n-parameter class of affine functions ${\stackrel{o}{u}}_J$:

$${\stackrel{o}{u}}_J(\stackrel{o}{x})=\sum _{\mu ,\nu =0}^n{J}_{\mu \nu }{x}_{\nu }{\stackrel{o}{\mathit{e}}}_{\mu }$$

(53)

where the matrix J is given by (41) and, as usual, ${\stackrel{o}{\mathit{e}}}_0=1$.

5. Conclusions

We derived the complete set of solutions to the generalized Cauchy–Riemann equations (2), supplemented by constraints (5), in the elliptic scator space of any dimension. Assuming merely local $C^2$ regularity, we determined all such scator holomorphic functions; they are in fact elementary and real-analytic everywhere. We note that only $C^1$ regularity is assumed in the definition of scator holomorphy, and it remains an open problem whether our result can be derived under this weaker assumption.

The obtained set of solutions is not very rich, but in the case of quaternionic analysis the analogous set is much smaller, consisting only of linear affine functions [10,11]. Therefore, in Clifford analysis, including quaternionic analysis, other definitions of holomorphicity are widely used, like Clifford-holomorphic or -monogenic functions, see, e.g., [12]. It would also be interesting to investigate such possibilities in the case of scator spaces.