Scator Holomorphic Functions

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 $b_0\ne 0$ (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. By the way, another example of a non-distributive algebra of hypercomplex numbers have been proposed recently [5].

In the hyperbolic case scators have potential physical interpretation and applications related to some deformations of the 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]), we focus on the most straightforward definition of holomorphicity, i.e., the existence of a direction-independent derivative at any point. The scator differentiability of this kind implies a system of partial differential equations [8], which can be considered as a generalization of the Cauchy-Riemann equations of the complex analysis:

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

(2)

for all $m\ne j$, where m and j take values from 1 to n. 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 get 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 been recently already solved [15], while for $n>2$ two classes of solutions have been earlier reported: linear affine functions [8] and components exponential functions [16,17]. We will show that, similarly as in the case $n=2$, there exists a third class of solutions.

2. 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.

2.1. The case ${\displaystyle \frac{\partial u_0}{\partial x_0}\ne 0}$.

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

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

(3)

where the right-hand sides depends 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:

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

(4)

where here and in the sequel we use the following notation

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

(5)

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

The compatibility conditions (4) can be easily reduced to the following form

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

(6)

The system (6) consists of partial differential equations for only one field, namely $u_0\equiv u$. Therefore, we plan first to solve the system (6), and then (knowing u) to integrate the linear equations of the first order for the remaining fields $u_j$ ($j=1,\dots ,n$).

2.1.1. Solving the Compatibility Conditions (6)

Substituting $u{,}_{jj}$ and $u{,}_{jm}$ from the first two equations of (6) into the last equation (6) we get

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

(7)

for $m,j=1,\dots ,n$ ($m\ne j$). An elementary computation yields

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

(8)

which is equivalent to

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

(9)

Hence $u{,}_{0m}u{,}_0=u{,}_{00}u{,}_m$ and

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

(10)

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

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

(11)

where $F_m$ are functions which do not depend on $x_0$. Hence also

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

(12)

including the case $m=j$. Now, we can simplify the system (6) using (11) and (12):

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

(13)

We point out that in the second equation four terms turned out to be identical ($u{,}_{00}{F}_m{F}_j$, up to a sign) and canceled out. In the third equation all terms by $u{,}_{00}$ canceled out, as well.

Finally, as a consequence of the generalized Cauchy-Riemann system (6) we get

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

(14)

which means that

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

(15)

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

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

(16)

where g and h are functions of n variables. The last equation (for $u{,}_j$) of (14) yields

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

(17)

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

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

(18)

and

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

(19)

which is equivalent to

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

(20)

Therefore

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

(21)

where $g_0$ is a constant, and, finally

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

(22)

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

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

(23)

Similarly as in the generic case, constraints on g and h follow from the last equation of (14):

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

(24)

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

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

(25)

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

2.1.2. Solving the Full System (3)

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

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

(26)

where $u{,}_0$ is given by (22). The first equation can be easily integrated

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

(27)

Inserting (27) into the third equation of (26) we get $D_j{,}_m=0$ (for $m\ne j$), i.e.,

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

(28)

Then the second equation of (26) reduces to

$$-u{,}_0{F}_j^2-u{F}_j{,}_j+D_j{,}_j=u{,}_0,\left[1ex\right]$$

(29)

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 (15) and (22)), we get

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

(30)

Thus

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

(31)

where $d_j$ are constants. Thus, by (27), we obtain

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

(32)

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

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

(33)

For ${\omega }_0=0$ everything simplifies: $u{,}_0=g_0$, $F_j=c_j$ and the system (26) reduces to

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

(34)

and its general solution is given by

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

(35)

In other words, the scator holomorphic function given by (25) and (35) is just the affine map

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

(36)

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

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

(37)

which actually is the Jacobi matrix of the map (36). While the symmetric form of the Jacobi matrix J is intriguing, an interpretation and implications of this phenomenon are not yet clear.

2.2. 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.

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

(38)

The last equation of (38) 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:

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

(39)

which means that

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

(40)

and, applying the first equation of (38), we get ${\rho }_m^{\prime \prime }=0$ and ${\psi }_m{,}_m=0$, which means that for any $m⩽n$ we have:

$$\begin{array}{c}u=c_1+c_0{x}_m,\hfill \\ u_m=-c_0{x}_0+{\psi }_m(x_1,\dots ,\widehat{x_m},\dots ,x_n),\hfill \\ u_j={\varphi }_{mj}(x_1,\dots ,\widehat{x_j},\dots ,x_n)(j\ne m),\hfill \end{array}$$

(41)

where $c_0$ and $c_1$ 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 by $\varphi$ 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. It is convenient to denote ${\varphi }_{mm}={\psi }_m$.

3. Final Results and Classification of Solutions

The results of previous section, expressed by the formulas (22), (33), (36) and (41), can be summarized by the following theorem.

Theorem 1.

The full set of solutions to the generalized Cauchy-Riemann equations (2) consists of three families.

• Components exponential functions

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

  (42)
  where ${\lambda }_0$, $a_{\mu }$ and $d_{\mu }$ ($\mu =0,1,\dots ,n$) are real constants.
• Affine functions of the special form:

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

  (43)
  where ${\lambda }_0$ and $d_{\mu }$ are real constants and $J_{\mu \nu }$ are entries of the constant matrix J given by (37), which is expressed by constants $c_{\mu }$ ($\mu =0,1,\dots ,n$).
• Exceptional solutions parameterized by $n^2$ arbitrary functions:

  $$\begin{array}{c}{u}_0=c_0{x}_m+c_1,\hfill \\ u_j=-c_0{x}_0{\delta }_{jm}+{\varphi }_{mj}(x_1,\dots ,\widehat{x_j},\dots ,x_n),\hfill \end{array}$$

  (44)

where $c_0$ and $c_1$ are real constants, ${\delta }_{mj}$ is Kronecker’s delta, ${\varphi }_{mj}$ are arbitrary functions of $n-1$ variables, and the notation $\widehat{x_j}$ means that ${\varphi }_{mj}$ depends on all variables $x_1,\dots ,x_n$ except $x_j$.

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 little bit different way, coincide with the results presented above. It should be noted that the paper [15] contains misprints in formulas (7), (32), and (33). In those formulas, the functions $V_1$ and $V_2$ should depend only on z while the functions $W_1$ and $W_2$ should depend only on y.

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:

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

(45)

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 arise from these symmetries. We use similar notation in both theorems to emphasize this relationship.

4. Conclusions

We derived the complete set of solutions to the generalized Cauchy-Riemann equations (2) in the elliptic scator space of any dimension. In other words, we found all scator holomorphic functions. 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 the Clifford analysis, including the quaternionic analysis, other definitions of holomorphicity are widely used, like Clifford-holomorphic or monogenic functions, see, e.g., [12]. It would be interesting to investigate such possibilities also in the case of the scator spaces.

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