# Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generalized Cauchy–Riemann System in the Case $\mathit{n}=\mathbf{2}$

**Theorem**

**1.**

- Components exponential functions$$\begin{array}{c}u={q}_{0}+{p}_{0}{e}^{{k}_{0}x}cos({k}_{0}y+{k}_{1})cos({k}_{0}z+{k}_{2})\phantom{\rule{4pt}{0ex}},\hfill \\ v={q}_{1}+{p}_{0}{e}^{{k}_{0}x}sin({k}_{0}y+{k}_{1})cos({k}_{0}z+{k}_{2})\phantom{\rule{4pt}{0ex}},\hfill \\ w={q}_{2}+{p}_{0}{e}^{{k}_{0}x}cos({k}_{0}y+{k}_{1})sin({k}_{0}z+{k}_{2})\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$$
- Linear functions$$\begin{array}{c}u={b}_{0}+{a}_{0}(x+{f}_{0}y+{g}_{0}z)\phantom{\rule{4pt}{0ex}},\hfill \\ v={b}_{1}+{a}_{0}(y-{f}_{0}x-{f}_{0}{g}_{0}z)\phantom{\rule{4pt}{0ex}},\hfill \\ w={b}_{2}+{a}_{0}(z-{g}_{0}x-{f}_{0}{g}_{0}y)\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$$
- Exceptional solutions$$\begin{array}{c}u={c}_{0}y+{c}_{1}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}v=-{c}_{0}x+{V}_{1}(z)\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}w={W}_{1}(z)\phantom{\rule{4pt}{0ex}},\hfill \\ u={d}_{0}z+{d}_{1}\phantom{\rule{4pt}{0ex}},\phantom{\rule{2.em}{0ex}}v={V}_{2}(y)\phantom{\rule{4pt}{0ex}},\phantom{\rule{2.em}{0ex}}w=-{d}_{0}x+{W}_{2}(z)\phantom{\rule{4pt}{0ex}},\hfill \end{array}$$

## 3. Components Exponential Functions

## 4. Linear Functions

## 5. Exceptional Solutions

- $u=u(y)$, $v=v(x,z)$, $w=w(x,y)$.Then ${w}_{x}=0$ and ${u}_{y}=-{v}_{x}$. Hence ${u}_{y}={c}_{0}=\mathrm{const}$ and we get the following solution:$$u={c}_{0}y+{c}_{1}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}v=-{c}_{0}x+{V}_{1}(z)\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}w={W}_{1}(y)\phantom{\rule{4pt}{0ex}},$$
- $u=u(z)$, $v=v(x,z)$, $w=w(x,y)$.Then, ${w}_{x}=-{u}_{z}$ and ${v}_{x}=0$. Hence ${u}_{z}={d}_{0}$ and, as a result, we get the solution:$$u={d}_{0}z+{d}_{1}\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}v={V}_{2}(y)\phantom{\rule{4pt}{0ex}},\phantom{\rule{1.em}{0ex}}w=-{d}_{0}x+{W}_{2}(z)\phantom{\rule{4pt}{0ex}},$$

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Fernández-Guasti, M.; Zaldívar, F. An elliptic non distributive algebra. Adv. Appl. Clifford Algebr.
**2013**, 23, 825–835. [Google Scholar] [CrossRef] - Kobus, A.; Cieśliński, J.L. On the Geometry of the Hyperbolic Scator Space in 1+2 Dimensions. Adv. Appl. Clifford Algebr.
**2017**, 27, 1369–1386. [Google Scholar] [CrossRef][Green Version] - Cieśliński, J.L.; Kobus, A. On the Product Rule for the Hyperbolic Scator Algebra. Axioms
**2020**, 9, 55. [Google Scholar] [CrossRef] - Fernández-Guasti, M. Time and space transformations in a scator deformed Lorentz metric. Eur. Phys. J. Plus
**2014**, 129, 195. [Google Scholar] [CrossRef] - Fernández-Guasti, M. Composition of velocities and momentum transformations in a scator-deformed Lorentz metric. Eur. Phys. J. Plus
**2020**, 135, 542. [Google Scholar] [CrossRef] - Fernández-Guasti, M. A Non-distributive Extension of Complex Numbers to Higher Dimensions. Adv. Appl. Clifford Algebr.
**2015**, 25, 829–849. [Google Scholar] [CrossRef] - Fernández-Guasti, M. Differential quotients in elliptic scator algebra. Math. Meth. Appl. Sci.
**2018**, 41, 4827–4840. [Google Scholar] [CrossRef] - Fernández-Guasti, M. Components exponential scator holomorphic function. Math. Meth. Appl. Sci.
**2020**, 43, 1017–1034. [Google Scholar] [CrossRef] - Sudbery, A. Quaternionic analysis. Math. Proc. Camb. Philos. Soc.
**1979**, 85, 199–225. [Google Scholar] [CrossRef][Green Version] - De Leo, S.; Rotelli, P.P. Quaternionic Analyticity. Appl. Math. Lett.
**2003**, 16, 1077–1081. [Google Scholar] [CrossRef] - Ryan, J. Clifford analysis. In Lectures on Clifford (Geometric) Algebras and Applications; Abłamowicz, R., Sobczyk, G., Eds.; Birkhäuser, Boston-Basel-Berlin: Basel, Switzerland, 2004; pp. 53–89. [Google Scholar]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cieśliński, J.L.; Zhalukevich, D.
Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case. *Symmetry* **2020**, *12*, 1550.
https://doi.org/10.3390/sym12091550

**AMA Style**

Cieśliński JL, Zhalukevich D.
Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case. *Symmetry*. 2020; 12(9):1550.
https://doi.org/10.3390/sym12091550

**Chicago/Turabian Style**

Cieśliński, Jan L., and Dzianis Zhalukevich.
2020. "Explicit Formulas for All Scator Holomorphic Functions in the (1+2)-Dimensional Case" *Symmetry* 12, no. 9: 1550.
https://doi.org/10.3390/sym12091550