# Static Potential Vector Fields in $\mathbb{R}^{4}$
and New Generalizations of the
Cauchy-Riemann System

Dmitry Bryukhov  https://orcid.org/0000-0002-8977-3282
Science City Fryazino, Russian Federation<EMAIL_ADDRESS>
###### Abstract.

This paper extends approach of recent author’s paper devoted to special
classes of exact solutions of the static Maxwell system in inhomogeneous
isotropic media and new generalizations of the Cauchy-Riemann system in
$\mathbb{R}^{3}$. Two families of generalizations of the Cauchy-Riemann system
with variable coefficients in $\mathbb{R}^{4}$ are presented in the context of
_Non-Euclidean geometry_. Analytic models of a wide range of static potential
meridional vector fields in $\mathbb{R}^{4}$ are characterized using a family
of Vekua type systems in cylindrical coordinates. The specifics of potential
meridional fields allows us to introduce the concept of four-dimensional
$\alpha$-meridional mappings of the first and second kind depending on the
values of a real parameter $\alpha$. In case $\alpha=2$ tools of the radially
holomorphic potential in $\mathbb{R}^{4}$ are developed in the context of
_Generalized axially symmetric potential theory (GASPT)_. Analytic models of
potential meridional fields in $\mathbb{R}^{4}$ generated by Bessel functions
of the first kind of integer order and quaternionic argument are described in
case $\alpha=2$ using radially anti-holomorphic functions in $\mathbb{R}^{4}$.
In case $\alpha=0$ the geometric specifics of four-dimensional harmonic
meridional mappings of the second kind is demonstrated in the context of the
theory of _Gradient dynamical systems with harmonic potential_.

###### Key words and phrases:

Generalizations of the Cauchy-Riemann system, Potential meridional fields in
$\mathbb{R}^{4}$, Four-dimensional $\alpha$-meridional mappings of the second
kind, The radially holomorphic potential in $\mathbb{R}^{4}$, Gradient
dynamical systems

###### 1991 Mathematics Subject Classification:

Primary 30G35, 35J46; Secondary 30C65, 35Q99, 34A34

##  1\. Introduction, Preliminaries, and Notation

###  1.1. Introduction

A rich variety of analytic models of potential vector fields
$\vec{V}=(V_{0},V_{1},V_{2},V_{3})$ in
$\mathbb{R}^{4}=\\{(x_{0},x_{1},x_{2},x_{3})\\}$ is described by the following
first-order static system including a variable $C^{1}$-coefficient
$\phi=\phi(x_{0},x_{1},x_{2},x_{3})>0$:

$\displaystyle\begin{cases}\mathrm{div}\,(\phi\ \vec{V})=0,\\\\[4.30554pt]
\frac{\partial{V_{0}}}{\partial{x_{1}}}=\frac{\partial{V_{1}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{V_{1}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{1}}},\quad\frac{\partial{V_{1}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{1}}},\quad\frac{\partial{V_{2}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{2}}}.\end{cases}$
(1.1)

In our setting the Euclidean space $\mathbb{R}^{4}$ involves the longitudinal
variable $x_{0}$ and the cylindrical radial variable
$\rho=\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$. The scalar potential
$h=h(x_{0},x_{1},x_{2},x_{3})$ in simply connected open domains
$\Lambda\subset\mathbb{R}^{4}$, where $\vec{V}=\mathrm{grad}\ h$, allows us to
reduce every $C^{1}$-solution of the system (1.1) to a $C^{2}$-solution of the
continuity equation

$\displaystyle\mathrm{div}\,(\phi\ \mathrm{grad}{\ h})=0.$ (1.2)

The space $\mathbb{R}^{4}$ and the coefficient
$\phi=\phi(x_{0},x_{1},x_{2},x_{3})$ may be interpreted as the phase space $X$
(or the state space) and the Jacobi last multiplier
$M=M(x_{0},x_{1},x_{2},x_{3})$, respectively, in the context of the theory of
_Gradient dynamical systems_ , where
$\vec{V}=\frac{d{\vec{x}}}{dt}=\mathrm{grad}\ h$ (see, e.g., [46, 35, 51, 36,
55, 47]).

Some types of harmonic potential vector fields have been extensively
investigated by Brackx, Delanghe, Sommen et al. in the context of
_Quaternionic analysis_ and the theory of _Quaternion-valued monogenic
functions_ since the 1970s (see, e.g., [58, 5, 18, 45, 6, 19]). Remarkable
extensions of the basic concepts of power series, holomorphicity, elementary
and special functions have been developed for the past 40 years, in
particular, in the context of _Quaternionic analysis and its modifications_
(see, e.g., [41, 34, 42, 24, 43, 44]), of the theory of _Holomorphic functions
in $n$-dimensional space_ (see, e.g., [32, 33]), of the theory of _Cullen
regular (also referred to as “slice regular”, “slice monogenic” or “slice
hyperholomorphic” in various settings) functions_ (see, e.g., [27, 13, 14, 15,
57, 28, 16, 26, 29].

Essentially new tools of quaternionic power series with quaternion-valued
coefficients have been developed by Leutwiler and Hempfling during the 1990s
to investigate analytic properties of exact solutions of the system

$\displaystyle(H_{4})\begin{cases}x_{3}\left(\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{1}}}{\partial{x_{1}}}-\frac{\partial{u_{2}}}{\partial{x_{2}}}-\frac{\partial{u_{3}}}{\partial{x_{3}}}\right)+2u_{3}=0\\\\[4.30554pt]
\frac{\partial{u_{0}}}{\partial{x_{1}}}=-\frac{\partial{u_{1}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{2}}}=-\frac{\partial{u_{2}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{3}}}=-\frac{\partial{u_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{u_{1}}}{\partial{x_{2}}}=\ \
\frac{\partial{u_{2}}}{\partial{x_{1}}},\quad\frac{\partial{u_{1}}}{\partial{x_{3}}}=\
\
\frac{\partial{u_{3}}}{\partial{x_{1}}},\quad\frac{\partial{u_{2}}}{\partial{x_{3}}}=\
\ \frac{\partial{u_{3}}}{\partial{x_{2}}}\end{cases}$

in the context of _Modified quaternionic analysis in $\mathbb{R}^{4}$_, where
$x_{3}>0$ (see, e.g., [41, 34, 42]). This system is called the system
$(H_{4})$ in honor of Hodge.

General class of exact solutions of the system $(H_{4})$ may be equivalently
represented as general class of exact solutions of the static system

$\displaystyle\begin{cases}\mathrm{div}\,(x_{3}^{-2}\
\vec{V})=0,\\\\[4.30554pt]
\frac{\partial{V_{0}}}{\partial{x_{1}}}=\frac{\partial{V_{1}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{V_{1}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{1}}},\quad\frac{\partial{V_{1}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{1}}},\quad\frac{\partial{V_{2}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{2}}},\end{cases}$
(1.3)

where $\phi=\phi(x_{3})=x_{3}^{-2}$; $\
(V_{0},V_{1},V_{2},V_{3})=(u_{0},-u_{1},-u_{2},-u_{3})$.

Assume that $\rho>0$. Axially symmetric Fueter’s construction in
$\mathbb{R}^{4}$ (see, e.g., [25, 41])

$\displaystyle F=F(x)=u_{0}+iu_{1}+ju_{2}+ku_{3}=u_{0}(x_{0},\rho)+I\
u_{\rho}(x_{0},\rho),$ $\displaystyle x=x_{0}+I\rho,\quad
I=\frac{ix_{1}+jx_{2}+kx_{3}}{\rho},\quad i^{2}=j^{2}=k^{2}=I^{2}=-1,$ (1.4)

was characterized by the author in 2003 [8] as joint class of analytic
solutions of the system $(H_{4})$ and axially symmetric generalization of the
Cauchy-Riemann system

$\displaystyle\begin{cases}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})\left(\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{1}}}{\partial{x_{1}}}-\frac{\partial{u_{2}}}{\partial{x_{2}}}-\frac{\partial{u_{3}}}{\partial{x_{3}}}\right)+2(x_{1}u_{1}+x_{2}u_{2}+x_{3}u_{3})=0\\\\[4.30554pt]
\frac{\partial{u_{0}}}{\partial{x_{1}}}=-\frac{\partial{u_{1}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{2}}}=-\frac{\partial{u_{2}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{3}}}=-\frac{\partial{u_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{u_{1}}}{\partial{x_{2}}}=\ \
\frac{\partial{u_{2}}}{\partial{x_{1}}},\quad\frac{\partial{u_{1}}}{\partial{x_{3}}}=\
\
\frac{\partial{u_{3}}}{\partial{x_{1}}},\quad\frac{\partial{u_{2}}}{\partial{x_{3}}}=\
\ \frac{\partial{u_{3}}}{\partial{x_{2}}}\end{cases}$ (1.5)

under conditions of

${u_{1}}{x_{2}}={u_{2}}{x_{1}},\quad{u_{1}}{x_{3}}={u_{3}}{x_{1}},\quad{u_{2}}{x_{3}}={u_{3}}{x_{2}},\quad
x_{3}>0.$ (1.6)

General class of exact solutions of the static system

$\displaystyle\begin{cases}\mathrm{div}\,(\rho^{-2}\
\vec{V})=0,\\\\[4.30554pt]
\frac{\partial{V_{0}}}{\partial{x_{1}}}=\frac{\partial{V_{1}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{V_{1}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{1}}},\quad\frac{\partial{V_{1}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{1}}},\quad\frac{\partial{V_{2}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{2}}},\end{cases}$
(1.7)

is equivalently represented as general class of exact solutions of the system
(1.5), where $\phi=\phi(\rho)=\rho^{-2}$; $\
(V_{0},V_{1},V_{2},V_{3})=(u_{0},-u_{1},-u_{2},-u_{3})$.

Surprisingly, applications of the system $(H_{4})$, the system (1.5) and
Fueter’s construction in $\mathbb{R}^{4}$ (1.4) in accordance with the systems
(1.3), (1.7) have been missed.

The main goal of this paper is to develop new tools of the theory of _Static
potential meridional vector fields in $\mathbb{R}^{4}$_ by means of different
generalizations of the Cauchy-Riemann system with variable coefficients.

The paper is organized as follows. In Section 2, two families of
generalizations of the Cauchy-Riemann system with variable coefficients in
$\mathbb{R}^{4}$ are provided. Analytic properties of potential meridional
fields are considered in the context of _Modified quaternionic analysis in
$\mathbb{R}^{4}$_, _Hyperbolic function theory in the skew-field of
quaternions_ and the theory of _Modified harmonic functions in
$\mathbb{R}^{4}$_. In Section 3, new concept of four-dimensional
$\alpha$-meridional mappings of the first and second kind, where
$\alpha\in\mathbb{R}$, is introduced. In Section 4, in case $\alpha=2$ tools
of the radially holomorphic potential in $\mathbb{R}^{4}$ are developed using
the concept of radially holomorphic functions in $\mathbb{R}^{4}$ introduced
by Gürlebeck, Habetha, Sprößig in the context of the theory of _Holomorphic
functions in n-dimensional space_. In Section 5, new analytic models of
potential meridional fields in $\mathbb{R}^{4}$ generated by the quaternionic
Fourier-Fueter cosine and sine transforms of real-valued original functions
are described in case $\alpha=2$ using radially anti-holomorphic functions in
$\mathbb{R}^{4}$. As a corollary, integral representations of Bessel functions
of the first kind of integer order and quaternionic argument are obtained. In
Section 6, in case $\alpha=0$ the geometric specifics of four-dimensional
harmonic meridional mappings of the second kind is demonstrated explicitly in
the context of _Stability theory of gradient dynamical systems with harmonic
potential_.

###  1.2. Preliminaries

New families of generalizations of the Cauchy-Riemann system with variable
coefficients in $\mathbb{R}^{4}$ may be provided by means of the following
first-order system:

$\displaystyle\begin{cases}\phi\left(\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{1}}}{\partial{x_{1}}}-\frac{\partial{u_{2}}}{\partial{x_{2}}}-\frac{\partial{u_{3}}}{\partial{x_{3}}}\right)+\left(\frac{\partial{\phi}}{\partial{x_{0}}}u_{0}-\frac{\partial{\phi}}{\partial{x_{1}}}u_{1}-\frac{\partial{\phi}}{\partial{x_{2}}}u_{2}-\frac{\partial{\phi}}{\partial{x_{3}}}u_{3}\right)=0,\\\\[4.30554pt]
\frac{\partial{u_{0}}}{\partial{x_{1}}}=-\frac{\partial{u_{1}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{2}}}=-\frac{\partial{u_{2}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{3}}}=-\frac{\partial{u_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{u_{1}}}{\partial{x_{2}}}=\ \
\frac{\partial{u_{2}}}{\partial{x_{1}}},\quad\frac{\partial{u_{1}}}{\partial{x_{3}}}=\
\
\frac{\partial{u_{3}}}{\partial{x_{1}}},\quad\frac{\partial{u_{2}}}{\partial{x_{3}}}=\
\ \frac{\partial{u_{3}}}{\partial{x_{2}}}.\end{cases}$ (1.8)

Suppose that $(V_{0},V_{1},V_{2},V_{3})=(u_{0},-u_{1},-u_{2},-u_{3})$. General
class of $C^{1}$-solutions of the system (1.1) may be equivalently represented
as general class of $C^{1}$-solutions of the system (1.8).

The continuity equation (1.2) in the expanded form is expressed as

$\displaystyle\phi\Delta
h+\frac{\partial{\phi}}{\partial{x_{0}}}\frac{\partial{h}}{\partial{x_{0}}}+\frac{\partial{\phi}}{\partial{x_{1}}}\frac{\partial{h}}{\partial{x_{1}}}+\frac{\partial{\phi}}{\partial{x_{2}}}\frac{\partial{h}}{\partial{x_{2}}}+\frac{\partial{\phi}}{\partial{x_{3}}}\frac{\partial{h}}{\partial{x_{3}}}=0,$
(1.9)

where the Laplacian $\
\Delta:=\frac{{\partial}^{2}{}}{{\partial{x_{0}}}^{2}}+\frac{{\partial}^{2}{}}{{\partial{x_{1}}}^{2}}+\frac{{\partial}^{2}{}}{{\partial{x_{2}}}^{2}}+\frac{{\partial}^{2}{}}{{\partial{x_{3}}}^{2}}$.

The equation

$h(x_{0},x_{1},x_{2},x_{3})=C=const,$ (1.10)

allows us to establish important properties of the equipotential hypersurfaces
in simply connected open domains $\Lambda\subset\mathbb{R}^{4}$. Using the
total differential $dh$, Eq. (1.10) is reformulated as an exact differential
equation (see, e.g., [60])

$\displaystyle
dh=\frac{\partial{h}}{\partial{x_{0}}}d{x_{0}}+\frac{\partial{h}}{\partial{x_{1}}}d{x_{1}}+\frac{\partial{h}}{\partial{x_{2}}}d{x_{2}}+\frac{\partial{h}}{\partial{x_{3}}}d{x_{3}}=0.$

Let $\varsigma$ be a real independent variable. Assume that homogeneous first-
order partial differential equation

$\frac{\partial{h}}{\partial{x_{0}}}W_{0}+\frac{\partial{h}}{\partial{x_{1}}}W_{1}+\frac{\partial{h}}{\partial{x_{2}}}W_{2}+\frac{\partial{h}}{\partial{x_{3}}}W_{3}=0$
(1.11)

is satisfied in $\Lambda$ such that

$W_{l}(x_{0},x_{1},x_{2},x_{3})=\frac{dx_{l}}{d\varsigma}\quad(l=0,1,2,3).$
(1.12)

The system $(\ref{characteristics-4})$ allows us to introduce the
characteristic vector field $\vec{W}=(W_{0},W_{1},W_{2},W_{3})$ for equation
(1.11) in $\Lambda$ in the context of the _Geometrical methods of the theory
of ordinary differential equations_ (see, e.g., [3]). The scalar potential
$h=h(x_{0},x_{1},x_{2},x_{3})$ is accordingly referred to as a first integral
of the characteristic vector field $\vec{W}=(W_{0},W_{1},W_{2},W_{3})$ (or of
its associated system
$\frac{dx_{0}}{W_{0}}=\frac{dx_{1}}{W_{1}}=\frac{dx_{2}}{W_{2}}=\frac{dx_{3}}{W_{3}}$)
in $\Lambda$ if and only if the equation (1.11) is satisfied in $\Lambda$
(see, e.g., [64, 3, 47]).

The equation (1.11) is geometrically characterized as the orthogonality
condition for vector fields $\vec{V}$ and $\vec{W}$:

$\displaystyle(\vec{V},\vec{W})=(\mathrm{grad}\ h,\vec{W})=0.$ (1.13)

Equation (1.13) is satisfied, in particular, under condition of
$\vec{V}=\mathrm{grad}\ h$ $=(u_{0},-u_{1},-u_{2},-u_{3})=0$.

######  Definition 1.1.

Let $\Lambda\subset\mathbb{R}^{4}$ be a simply connected open domain. Every
point $x^{*}=(x_{0}^{*},x_{1}^{*},x_{2}^{*},x_{3}^{*})\in\Lambda$ under
condition of $\mathrm{grad}\ h(x^{*})=0$ is called a critical point of the
scalar potential $h$ in $\Lambda$. The set of critical points is called the
critical set of $h$ in $\Lambda$.

Geometric and topological properties of the critical sets of the scalar
potential $h$ are of particular interest to _Catastrophe theory_ (see, e.g.,
[30]).

The Hessian matrix $\mathbf{H}(h(x))$ of the scalar potential $h$ may be
considered as the Jacobian matrix $\mathbf{J}(\vec{V}(x))$ of the vector field
$\vec{V}$, where $J_{lm}=\frac{\partial{V_{l}}}{\partial{x_{m}}}\ $
$(l,m=0,1,2,3)$:

$\left(\begin{array}[]{rrrr}\frac{\partial{V_{0}}}{\partial{x_{0}}}&\frac{\partial{V_{0}}}{\partial{x_{1}}}&\frac{\partial{V_{0}}}{\partial{x_{2}}}&\frac{\partial{V_{0}}}{\partial{x_{3}}}\\\\[4.30554pt]
\frac{\partial{V_{1}}}{\partial{x_{0}}}&\frac{\partial{V_{1}}}{\partial{x_{1}}}&\frac{\partial{V_{1}}}{\partial{x_{2}}}&\frac{\partial{V_{1}}}{\partial{x_{3}}}\\\\[4.30554pt]
\frac{\partial{V_{2}}}{\partial{x_{0}}}&\frac{\partial{V_{2}}}{\partial{x_{1}}}&\frac{\partial{V_{2}}}{\partial{x_{2}}}&\frac{\partial{V_{2}}}{\partial{x_{3}}}\\\\[4.30554pt]
\frac{\partial{V_{3}}}{\partial{x_{0}}}&\frac{\partial{V_{3}}}{\partial{x_{1}}}&\frac{\partial{V_{3}}}{\partial{x_{2}}}&\frac{\partial{V_{3}}}{\partial{x_{3}}}\end{array}\right)=\left(\begin{array}[]{rrrr}\
\ \frac{\partial{u_{0}}}{\partial{x_{0}}}&\ \
\frac{\partial{u_{0}}}{\partial{x_{1}}}&\ \
\frac{\partial{u_{0}}}{\partial{x_{2}}}&\ \
\frac{\partial{u_{0}}}{\partial{x_{2}}}\\\\[4.30554pt]
-\frac{\partial{u_{1}}}{\partial{x_{0}}}&-\frac{\partial{u_{1}}}{\partial{x_{1}}}&-\frac{\partial{u_{1}}}{\partial{x_{2}}}&-\frac{\partial{u_{1}}}{\partial{x_{2}}}\\\\[4.30554pt]
-\frac{\partial{u_{2}}}{\partial{x_{0}}}&-\frac{\partial{u_{2}}}{\partial{x_{1}}}&-\frac{\partial{u_{2}}}{\partial{x_{2}}}&-\frac{\partial{u_{2}}}{\partial{x_{2}}}\\\\[4.30554pt]
-\frac{\partial{u_{3}}}{\partial{x_{0}}}&-\frac{\partial{u_{3}}}{\partial{x_{1}}}&-\frac{\partial{u_{3}}}{\partial{x_{2}}}&-\frac{\partial{u_{3}}}{\partial{x_{2}}}\end{array}\right)$
(1.14)

The characteristic equation of the Hessian matrix $\mathbf{H}(h(x))$ in the
general four-dimensional setting is expressed as

$\displaystyle\lambda^{4}-I_{\mathbf{J}(\vec{V})}\lambda^{3}+II_{\mathbf{J}(\vec{V})}\lambda^{2}-III_{\mathbf{J}(\vec{V})}\lambda+IV_{\mathbf{J}(\vec{V})}=0.$
(1.15)

The principal invariants $I_{\mathbf{J}(\vec{V})}$,
$II_{\mathbf{J}(\vec{V})}$, $III_{\mathbf{J}(\vec{V})}$,
$IV_{\mathbf{J}(\vec{V})}$ of the Jacobian matrix (1.14) are given by formulas

$\displaystyle\begin{cases}I_{\mathbf{J}(\vec{V})}=\lambda_{0}+\lambda_{1}+\lambda_{2}+\lambda_{3}=J_{00}+J_{11}+J_{22}+J_{33},\\\\[4.30554pt]
II_{\mathbf{J}(\vec{V})}=\lambda_{0}\lambda_{1}+\lambda_{0}\lambda_{2}+\lambda_{0}\lambda_{3}+\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{3}=\\\\[4.30554pt]
J_{00}J_{11}+J_{00}J_{22}+J_{00}J_{33}+J_{11}J_{22}+J_{11}J_{33}+J_{22}J_{33},\\\\[4.30554pt]
III_{\mathbf{J}(\vec{V})}=\lambda_{0}\lambda_{1}\lambda_{2}+\lambda_{0}\lambda_{1}\lambda_{3}+\lambda_{0}\lambda_{2}\lambda_{3}+\lambda_{1}\lambda_{2}\lambda_{3}=\\\\[4.30554pt]
J_{00}J_{11}J_{22}+J_{00}J_{11}J_{33}+J_{00}J_{22}J_{33}+J_{11}J_{22}J_{33}\\\\[4.30554pt]
+2J_{01}J_{02}J_{12}+2J_{01}J_{03}J_{13}+2J_{02}J_{03}J_{23}+2J_{12}J_{13}J_{23}\\\\[4.30554pt]
-J_{00}(J_{12})^{2}-J_{00}(J_{13})^{2}-J_{00}(J_{23})^{2}-J_{11}(J_{02})^{2}-J_{11}(J_{03})^{2}-J_{11}(J_{23})^{2}\\\\[4.30554pt]
-J_{22}(J_{01})^{2}-J_{22}(J_{03})^{2}-J_{22}(J_{13})^{2}-J_{33}(J_{01})^{2}-J_{33}(J_{02})^{2}-J_{33}(J_{12})^{2},\\\\[4.30554pt]
IV_{\mathbf{J}(\vec{V})}=\lambda_{0}\lambda_{1}\lambda_{2}\lambda_{3}=J_{00}J_{11}J_{22}J_{33}+2J_{00}J_{12}J_{13}J_{23}+2J_{01}J_{02}J_{12}J_{33}\\\\[4.30554pt]
+2J_{02}J_{03}J_{23}J_{11}+(J_{01}J_{23})^{2}+2J_{01}J_{03}J_{13}J_{22}+(J_{02}J_{13})^{2}+(J_{03}J_{12})^{2}\\\\[4.30554pt]
-2J_{01}J_{03}J_{12}J_{23}-2J_{01}J_{02}J_{13}J_{23}-2J_{02}J_{03}J_{12}J_{13}-J_{00}J_{11}(J_{23})^{2}-\\\\[4.30554pt]
J_{00}J_{22}(J_{13})^{2}-J_{00}J_{33}(J_{12})^{2}-J_{11}J_{22}(J_{03})^{2}-J_{11}J_{33}(J_{02})^{2}-J_{22}J_{33}(J_{01})^{2}.\end{cases}$

######  Definition 1.2.

Every point $x\in\Lambda$ under condition of $\det\mathbf{J}(\vec{V}(x))=0$ is
called a degenerate point of the Jacobian matrix $\mathbf{J}(\vec{V}(x))$ in
$\Lambda$.

Meanwhile, static potential vector fields in $\mathbb{R}^{4}$ may be
investigated in the context of _Non-Euclidean geometry_ using the Laplace-
Beltrami equation (see, e.g., [1, 41, 34, 9, 10])

$\displaystyle\Delta_{B}\ h:=\phi^{-2}\mathrm{div}(\phi\ \mathrm{grad}{\
h})=0$

with respect to the conformal metric

$\displaystyle
ds^{2}=\phi(d{x_{0}}^{2}+d{x_{1}}^{2}+d{x_{2}}^{2}+d{x_{3}}^{2}).$ (1.16)

Euclidean geometry is provided in case $\phi=const$. Harmonic potential fields
$\vec{V}$ in $\mathbb{R}^{4}$ satisfy the first-order static system

$\displaystyle\begin{cases}\frac{\partial{V_{0}}}{\partial{x_{0}}}+\frac{\partial{V_{1}}}{\partial{x_{1}}}+\frac{\partial{V_{2}}}{\partial{x_{2}}}+\frac{\partial{V_{3}}}{\partial{x_{3}}}=0,\\\\[4.30554pt]
\frac{\partial{V_{0}}}{\partial{x_{1}}}=\frac{\partial{V_{1}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{V_{1}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{1}}},\quad\frac{\partial{V_{1}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{1}}},\quad\frac{\partial{V_{2}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{2}}}.\end{cases}$

This system is well-known in the context of _Fourier analysis on Euclidean
spaces_ (see, e.g., [56]). On the other hand, harmonic potential fields in
$\mathbb{R}^{4}$ may be equivalently represented as general class of of
analytic solutions of the system

$\displaystyle(R_{4})\begin{cases}\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{1}}}{\partial{x_{1}}}-\frac{\partial{u_{2}}}{\partial{x_{2}}}-\frac{\partial{u_{3}}}{\partial{x_{3}}}=0,\\\\[4.30554pt]
\frac{\partial{u_{0}}}{\partial{x_{1}}}=-\frac{\partial{u_{1}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{2}}}=-\frac{\partial{u_{2}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{3}}}=-\frac{\partial{u_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{u_{1}}}{\partial{x_{2}}}=\ \
\frac{\partial{u_{2}}}{\partial{x_{1}}},\quad\frac{\partial{u_{1}}}{\partial{x_{3}}}=\
\
\frac{\partial{u_{3}}}{\partial{x_{1}}},\quad\frac{\partial{u_{2}}}{\partial{x_{3}}}=\
\ \frac{\partial{u_{3}}}{\partial{x_{2}}},\end{cases}$

where $(u_{0},u_{1},u_{2},u_{3}):=(V_{0},-V_{1},-V_{2},-V_{3})$. This system
is called the system $(R_{4})$ in honor of Riesz. Four-dimensional harmonic
mappings $u=u_{0}+iu_{1}+ju_{2}+ju_{3}:\Lambda\rightarrow\mathbb{R}^{4}$ in
the context of _Quaternionic analysis_ are referred to as quaternion-valued
monogenic functions (see, e.g., [6, 42, 19]).

The system (1.8) in the context of _Modified quaternionic analysis in
$\mathbb{R}^{4}$_ (see, e.g., [41, 34, 42]), _Hyperbolic function theory in
the skew-field of quaternions_ (see, e.g., [24]) and the theory of _Modified
harmonic functions in $\mathbb{R}^{4}$_ (see, e.g., [43, 44]) may be
characterized as generalized non-Euclidean modification of the system
$(R_{4})$ with respect to the conformal metric (1.16).

###  1.3. Notation

The real algebra of quaternions $\mathbb{H}$ is a four dimensional skew
algebra over the real field generated by real unity $1$. Three imaginary
unities $i,j,$ and $k$ satisfy to multiplication rules

$\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1,\quad ij=-ji=k.$

The independent quaternionic variable is defined as

$x=x_{0}+ix_{1}+jx_{2}+kx_{3}.$

The quaternion conjugation of $x$ is defined by the following automorphism:

$x\mapsto\overline{x}:=x_{0}-ix_{1}-jx_{2}-kx_{3}.$

In such way, we deal with the Euclidean norm in $\mathbb{R}^{4}$

$\|x\|^{2}:=x\overline{x}=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}:=r^{2},$

and the identification

$x=x_{0}+ix_{1}+jx_{2}+kx_{3}\sim(x_{0},x_{1},x_{2},x_{3})$

between $\mathbb{H}$ and $\mathbb{R}^{4}$ is valid. Moreover, for every non-
zero value of $x$ an unique inverse value exists:
$x^{-1}=\overline{x}/\|x\|^{2}.$

The dependent quaternionic variable is defined as

$u=u_{0}+iu_{1}+ju_{2}+ju_{3}\sim(u_{0},u_{1},u_{2},u_{3}).$

The quaternion conjugation of $u$ is defined by the following automorphism:

$u\mapsto\overline{u}:=u_{0}-iu_{1}-ju_{2}-ku_{3}.$

Assume that $x_{3}>0$. In cylindrical coordinates in $\mathbb{R}^{4}$ we
obtain

$x=x_{0}+\rho(i\cos{\theta}+j\sin{\theta}\cos{\psi}+k\sin{\theta}\sin{\psi}),$
where

$x_{1}=\rho\cos{\theta},\quad x_{2}=\rho\sin{\theta}\cos{\psi},\quad
x_{3}=\rho\sin{\theta}\sin{\psi},$

$\varphi=\arccos\frac{x_{0}}{r}\ \
(0<\varphi<\pi),\quad\theta=\arccos\frac{x_{1}}{\rho}\ \ (0\leq\theta\leq
2\pi),$

$\psi=\mathrm{arccot}\frac{x_{2}}{x_{3}}\ \ (0<\psi<\pi).$

######  Definition 1.3.

Let $\Omega\subset\mathbb{R}^{4}$ be an open set. Every continuously
differentiable mapping
$u=u_{0}+iu_{1}+ju_{2}+ju_{3}:\Omega\rightarrow\mathbb{R}^{4}$ is called
quaternion-valued $C^{1}$-function in $\Omega$.

##  2\. Two Types of Potential Vector Fields in $\mathbb{R}^{4}$ and
Criterions of Potential Meridional Fields

To provide the first type of potential vector fields in $\mathbb{R}^{4}$,
assume that $C^{1}$-coefficient $\phi(x_{0},x_{1},x_{2},x_{3})$ depends only
on variable $x_{3}$ so that $\phi=\phi(x_{3})>0$. The system $(\ref{Bryukhov-
general-4})$ is described as

$\displaystyle\begin{cases}\phi(x_{3})\left(\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{1}}}{\partial{x_{1}}}-\frac{\partial{u_{2}}}{\partial{x_{2}}}-\frac{\partial{u_{3}}}{\partial{x_{3}}}\right)-\frac{d{\phi}}{d{x_{3}}}u_{3}=0,\\\\[4.30554pt]
\frac{\partial{u_{0}}}{\partial{x_{1}}}=-\frac{\partial{u_{1}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{2}}}=-\frac{\partial{u_{2}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{3}}}=-\frac{\partial{u_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{u_{1}}}{\partial{x_{2}}}=\ \
\frac{\partial{u_{2}}}{\partial{x_{1}}},\quad\frac{\partial{u_{1}}}{\partial{x_{3}}}=\
\
\frac{\partial{u_{3}}}{\partial{x_{1}}},\quad\frac{\partial{u_{2}}}{\partial{x_{3}}}=\
\ \frac{\partial{u_{3}}}{\partial{x_{2}}}.\end{cases}$ (2.1)

New properties of analytic models of potential vector fields may be
investigated in more detail in case ${\phi}(x_{3})=x_{3}^{-\alpha}$
$(x_{3}>0$, $\alpha\in\mathbb{R})$. The static system (1.1) is expressed as

$\displaystyle\begin{cases}x_{3}\mathrm{div}\ {\vec{V}}-\alpha
V_{3}=0,\\\\[4.30554pt]
\frac{\partial{V_{0}}}{\partial{x_{1}}}=\frac{\partial{V_{1}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{V_{1}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{1}}},\quad\frac{\partial{V_{1}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{1}}},\quad\frac{\partial{V_{2}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{2}}},\end{cases}$
(2.2)

and the system (2.1) is simplified:

$\displaystyle\begin{cases}x_{3}\left(\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{1}}}{\partial{x_{1}}}-\frac{\partial{u_{2}}}{\partial{x_{2}}}-\frac{\partial{u_{3}}}{\partial{x_{3}}}\right)+\alpha
u_{3}=0\\\\[4.30554pt]
\frac{\partial{u_{0}}}{\partial{x_{1}}}=-\frac{\partial{u_{1}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{2}}}=-\frac{\partial{u_{2}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{3}}}=-\frac{\partial{u_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{u_{1}}}{\partial{x_{2}}}=\ \
\frac{\partial{u_{2}}}{\partial{x_{1}}},\quad\frac{\partial{u_{1}}}{\partial{x_{3}}}=\
\
\frac{\partial{u_{3}}}{\partial{x_{1}}},\quad\frac{\partial{u_{2}}}{\partial{x_{3}}}=\
\ \frac{\partial{u_{3}}}{\partial{x_{2}}}.\end{cases}$ (2.3)

The system (2.3) was first introduced by Eriksson and Orelma in 2019 in the
context of the theory of _Hyperbolic function theory in the skew-field of
quaternions_ [24]. This system demonstrates explicitly a family of
generalizations of the Cauchy-Riemann system in accordance with the system
(2.2) for different values of the parameter $\alpha$.

When $\alpha>0$, the system (2.3) may be characterized as $\alpha$-hyperbolic
non-Euclidean modification of the system $(R)$ with respect to the conformal
metric defined on the halfspace $\\{x_{3}>0\\}$ by the formula

$\displaystyle
ds^{2}=\frac{d{x_{0}}^{2}+d{x_{1}}^{2}+d{x_{2}}^{2}+d{x_{3}}^{2}}{x_{2}^{\alpha}}.$

The continuity equation (1.9) takes the form of the Weinstein equation in
$\mathbb{R}^{4}$ for any value of the parameter $\alpha$ (see, e.g., [41, 34,
24, 43, 44])

$x_{3}\Delta{h}-\alpha\frac{\partial{h}}{\partial{x_{3}}}=0.$ (2.4)

Meanwhile, nowadays solutions of the Weinstein equation (2.4) in case
$\alpha=2$ in the context of _Hyperbolic function theory in the skew-field of
quaternions_ are referred to as 2-hyperbolic harmonic functions in
$\mathbb{R}^{4}$ [24]. The critical sets of 2-hyperbolic harmonic functions
$h=h(x_{0},x_{1},x_{2},x_{3})$ within Fueter’s construction in
$\mathbb{R}^{4}$ (1.4), where
$F=F(x)=\frac{\partial{h}}{\partial{x_{0}}}-i\frac{\partial{h}}{\partial{x_{1}}}-j\frac{\partial{h}}{\partial{x_{1}}}-k\frac{\partial{h}}{\partial{x_{1}}}$,
under conditions of $x_{1}\neq 0,x_{2}\neq 0,x_{3}\neq 0$ coincide with the
sets of zeros of $F=F(x)$ (on the structure of the sets of zeros of
quaternionic polynomials with real coefficients see, e.g., [52, 59]).

######  Definition 2.1.

Let $\Lambda\subset\mathbb{R}^{4}$ $(x_{3}>0)$ be a simply connected open
domain, $\alpha>0$. Every exact solution of Eq. (2.4) in $\mathbb{R}^{4}$ is
called $\alpha$-hyperbolic harmonic potential in $\mathbb{R}^{4}$.

When $\alpha=0$, the system (2.3) becomes the system $(R_{4})$, while the
Weinstein equation in $\mathbb{R}^{4}$ (2.4) becomes the Laplace equation.

When $\alpha<0$, solutions of Eq. (2.4) in the context of _Modified harmonic
functions in $\mathbb{R}^{4}$_ are referred to as $-\alpha$-modified harmonic
functions in $\mathbb{R}^{4}$ (see, e.g., [43, 44]). Properties of homogeneous
polynomial solutions of Eq. (2.4) in spherical coordinates on the unit half
sphere
$S_{+}=\\{(x_{0},x_{1},x_{2},x_{3}):x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1$,
$\ x_{3}>0\\}$ have been recently studied by Leutwiler.

To provide the second type of potential vector fields in $\mathbb{R}^{4}$,
assume that $C^{1}$-coefficient $\phi(x_{0},x_{1},x_{2},x_{3})$ depends only
on the cylindrical radial variable $\rho$ so that $\phi=\phi(\rho)>0$. The
system (1.8) is described as

$\displaystyle\begin{cases}\phi(\rho)\left(\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{1}}}{\partial{x_{1}}}-\frac{\partial{u_{2}}}{\partial{x_{2}}}-\frac{\partial{u_{3}}}{\partial{x_{3}}}\right)-\left(\frac{\partial{\phi(\rho)}}{\partial{x_{1}}}u_{1}+\frac{\partial{\phi(\rho)}}{\partial{x_{2}}}u_{2}+\frac{\partial{\phi(\rho)}}{\partial{x_{3}}}u_{3}\right)=0,\\\\[4.30554pt]
\frac{\partial{u_{0}}}{\partial{x_{1}}}=-\frac{\partial{u_{1}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{2}}}=-\frac{\partial{u_{2}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{3}}}=-\frac{\partial{u_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{u_{1}}}{\partial{x_{2}}}=\ \
\frac{\partial{u_{2}}}{\partial{x_{1}}},\quad\frac{\partial{u_{1}}}{\partial{x_{3}}}=\
\
\frac{\partial{u_{3}}}{\partial{x_{1}}},\quad\frac{\partial{u_{2}}}{\partial{x_{3}}}=\
\ \frac{\partial{u_{3}}}{\partial{x_{2}}}.\end{cases}$ (2.5)

New properties of axially symmetric analytic models of potential vector fields
may be investigated in more detail in case $\phi(\rho)=\rho^{-\alpha}$
$(\rho>0$, $\alpha\in\mathbb{R})$. The static system (1.1) is expressed as

$\displaystyle\begin{cases}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})\ \mathrm{div}\
{\vec{V}}-\alpha\left(x_{1}V_{1}+x_{2}V_{2}+x_{3}V_{3}\right)=0,\\\\[4.30554pt]
\frac{\partial{V_{0}}}{\partial{x_{1}}}=\frac{\partial{V_{1}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{0}}},\quad\frac{\partial{V_{0}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{V_{1}}}{\partial{x_{2}}}=\frac{\partial{V_{2}}}{\partial{x_{1}}},\quad\frac{\partial{V_{1}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{1}}},\quad\frac{\partial{V_{2}}}{\partial{x_{3}}}=\frac{\partial{V_{3}}}{\partial{x_{2}}},\end{cases}$
(2.6)

and the system (2.5) is simplified:

$\displaystyle\begin{cases}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})\left(\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{1}}}{\partial{x_{1}}}-\frac{\partial{u_{2}}}{\partial{x_{2}}}-\frac{\partial{u_{3}}}{\partial{x_{3}}}\right)+\alpha(x_{1}u_{1}+x_{2}u_{2}+x_{3}u_{3})=0\\\\[4.30554pt]
\frac{\partial{u_{0}}}{\partial{x_{1}}}=-\frac{\partial{u_{1}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{2}}}=-\frac{\partial{u_{2}}}{\partial{x_{0}}},\quad\frac{\partial{u_{0}}}{\partial{x_{3}}}=-\frac{\partial{u_{3}}}{\partial{x_{0}}},\\\\[4.30554pt]
\frac{\partial{u_{1}}}{\partial{x_{2}}}=\ \
\frac{\partial{u_{2}}}{\partial{x_{1}}},\quad\frac{\partial{u_{1}}}{\partial{x_{3}}}=\
\
\frac{\partial{u_{3}}}{\partial{x_{1}}},\quad\frac{\partial{u_{2}}}{\partial{x_{3}}}=\
\ \frac{\partial{u_{3}}}{\partial{x_{2}}}.\end{cases}$ (2.7)

This system demonstrates explicitly a family of axially symmetric
generalizations of the Cauchy-Riemann system in accordance with the system
(2.6) for different values of the parameter $\alpha$.

The continuity equation (1.9) is written as

$(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})\Delta{h}-\alpha\left(x_{1}\frac{\partial{h}}{\partial{x_{1}}}+x_{2}\frac{\partial{h}}{\partial{x_{2}}}+x_{3}\frac{\partial{h}}{\partial{x_{3}}}\right)=0.$
(2.8)

When $\alpha>0$, the system (2.7) may be characterized as $\alpha$-axial-
hyperbolic non-Euclidean modification of the system $(R_{4})$ with respect to
the conformal metric defined outside the axis $x_{0}$ by the formula

$ds^{2}=\frac{d{x_{0}}^{2}+d{x_{1}}^{2}+d{x_{2}}^{2}+d{x_{3}}^{2}}{\rho^{\alpha}}.$

######  Definition 2.2.

Let $\Lambda\subset\mathbb{R}^{4}$ $(\rho>0)$ be a simply connected open
domain, $\alpha>0$. Every exact solution of Eq. (2.8) in $\mathbb{R}^{4}$ is
called $\alpha$-axial-hyperbolic harmonic potential in $\mathbb{R}^{4}$.

Let us compare properties of $\alpha$-hyperbolic harmonic potentials and
$\alpha$-axial-hyperbolic harmonic potentials in $\mathbb{R}^{4}$ in Cartesian
coordinates. This immediately leads to the following formulation.

######  Proposition 2.3 (The first criterion).

Any $\alpha$-hyperbolic harmonic potential $h=h(x_{0},x_{1},x_{2},x_{3})$ in
$\Lambda\subset\mathbb{R}^{4}$ $(x_{3}>0)$ represents an $\alpha$-axial-
hyperbolic harmonic potential in $\Lambda$ if and only if
$x_{2}\frac{\partial{h}}{\partial{x_{1}}}=x_{1}\frac{\partial{h}}{\partial{x_{2}}},$
$x_{3}\frac{\partial{h}}{\partial{x_{1}}}=x_{1}\frac{\partial{h}}{\partial{x_{3}}},$
$x_{3}\frac{\partial{h}}{\partial{x_{2}}}=x_{2}\frac{\partial{h}}{\partial{x_{3}}}.$
Each condition $x_{m}=0$ $(m=1,2,3)$ within joint class of $\alpha$-hyperbolic
harmonic and $\alpha$-axial-hyperbolic harmonic potentials implies that the
component $u_{m}=\frac{\partial{h}}{\partial{x_{m}}}$ vanishes.

######  Remark 2.4.

Necessary and sufficient conditions of joint class of $\alpha$-hyperbolic
harmonic and $\alpha$-axial-hyperbolic harmonic potentials in $\mathbb{R}^{4}$
coincide with conditions (1.6) of joint class of analytic solutions of the
system $(H_{4})$ and the system (1.5).

Let us now compare properties of $\alpha$-hyperbolic harmonic potentials and
$\alpha$-axial-hyperbolic harmonic potentials in $\mathbb{R}^{4}$ in
cylindrical coordinates. Eq. (2.8) in cylindrical coordinates is written as

$\displaystyle\rho^{2}\left(\frac{\partial{{}^{2}}{h}}{\partial{x_{0}}^{2}}+\frac{\partial{{}^{2}}{h}}{\partial{\rho}^{2}}\right)-(\alpha-2)\rho\frac{\partial{h}}{\partial{\rho}}+\cot{\theta}\frac{\partial{h}}{\partial{\theta}}+\frac{\partial{{}^{2}}{h}}{\partial{\theta}^{2}}+\frac{1}{\sin^{2}\theta}\frac{\partial{{}^{2}}{h}}{\partial{\psi}^{2}}=0.$

The Weinstein equation in $\mathbb{R}^{4}$ $(\ref{alpha-hyperbolic-4})$ in
cylindrical coordinates takes the following form:

$\displaystyle\rho^{2}\left(\frac{\partial{{}^{2}}{h}}{\partial{x_{0}}^{2}}+\frac{\partial{{}^{2}}{h}}{\partial{\rho}^{2}}\right)+2\rho\frac{\partial{h}}{\partial{\rho}}+\cot{\theta}\frac{\partial{h}}{\partial{\theta}}+\frac{\partial{{}^{2}}{h}}{\partial{\theta}^{2}}+\frac{1}{\sin^{2}\theta}\frac{\partial{{}^{2}}{h}}{\partial{\psi}^{2}}$
$\displaystyle-\alpha\rho\frac{\partial{h}}{\partial{\rho}}-\alpha\cot{\theta}\frac{\partial{h}}{\partial{\theta}}-\alpha\frac{\cot{\psi}}{\sin^{2}{\theta}}\frac{\partial{h}}{\partial{\psi}}=0.$

This immediately leads to the following formulation.

######  Proposition 2.5 (The second criterion).

Every $\alpha$-hyperbolic harmonic potential $h=h(x_{0},x_{1},x_{2},x_{3})$ in
$\Lambda\subset\mathbb{R}^{4}$ $(x_{3}>0)$ represents an $\alpha$-axial-
hyperbolic harmonic potential in $\Lambda$ if and only if in cylindrical
coordinates
$\frac{\partial{h}}{\partial{\theta}}=0,\frac{\partial{h}}{\partial{\psi}}=0.$

######  Remark 2.6.

New approach may be efficiently developed in the context of the theory of
_Hyperbolic function theory in the skew-field of quaternions_ and the theory
of _Modified harmonic functions in $\mathbb{R}^{4}$_ under conditions of
$\frac{\partial{h}}{\partial{\theta}}=0$,
$\frac{\partial{h}}{\partial{\psi}}=0$.

It should be noted that the vector $\vec{V}=(V_{0},V_{1},V_{2},V_{3})$ within
potential meridional fields in $\mathbb{R}^{4}$ is independent of two angles
$\psi$ and $\theta$, herewith
$V_{\psi}:=\frac{\partial{h}}{\partial{\psi}}\equiv 0$ and
$V_{\theta}:=\frac{\partial{h}}{\partial{\theta}}\equiv 0$.

As it follows from the first and second criterions, new joint class of
$\alpha$-hyperbolic harmonic and $\alpha$-axial-hyperbolic harmonic potentials
in simply connected open domains $\Lambda\subset\mathbb{R}^{4}$ $(x_{3}>0)$
may be characterized as general class of potential meridional fields
$\vec{V}$, where $\phi(\rho)=\rho^{-\alpha}$. Every scalar potential $h$
within the joint class is independent of angles $\psi$, $\theta$ such that
$h(x_{0},\rho,\theta,\psi)$ $=g(x_{0},\rho),$
$V_{0}=\frac{\partial{g}}{\partial{x_{0}}},$
$V_{1}=\frac{\partial{g}}{\partial{\rho}}\frac{x_{1}}{\rho}$,
$V_{2}=\frac{\partial{g}}{\partial{\rho}}\frac{x_{2}}{\rho}$,
$V_{3}=\frac{\partial{g}}{\partial{\rho}}\frac{x_{3}}{\rho}$, herewith

$\displaystyle\rho\left(\frac{\partial{{}^{2}}{g}}{\partial{x_{0}}^{2}}+\frac{\partial{{}^{2}}{g}}{\partial{\rho}^{2}}\right)-(\alpha-2)\frac{\partial{g}}{\partial{\rho}}=0.$
(2.9)

Equation (2.9), where $\hat{k}=-(\alpha-2)$, is referred to as the elliptic
Euler-Poisson-Darboux type equation in cylindrical coordinates (see, e.g.,
[20, 2, 10]), or generalized axially symmetric potential equation _(GASPE)_ in
the context of _GASPT_ (see, e.g., [62, 37, 17, 65, 31]). Exact solutions
$g=g(x_{0},\rho)$ of Eq. (2.9) are often referred to as generalized axially
symmetric potentials.

Every exact solution of Eq. (2.9) as generalized axially symmetric potential
indicates the existence of the Stokes’ stream function
$\hat{g}=\hat{g}(x_{0},\rho)$, which is defined by the generalized Stokes-
Beltrami system in the meridian half-plane $(\rho>0)$ (see, e.g., [62, 10]):

$\left\\{\begin{array}[]{l}{\rho}^{2-\alpha}\frac{\partial{g}}{\partial{x_{0}}}=\frac{\partial{\hat{g}}}{\partial{\rho}},\\\
{\rho}^{2-\alpha}\frac{\partial{g}}{\partial{\rho}}=-\frac{\partial{\hat{g}}}{\partial{x_{0}}}.\end{array}\right.$
(2.10)

The Stokes’ stream function $\hat{g}=\hat{g}(x_{0},\rho)$, in contrast to
generalized axially symmetric potential $g=g(x_{0},\rho)$, satisfies the
following equation:

$\rho\left(\frac{\partial{{}^{2}}{\hat{g}}}{\partial{x_{0}}^{2}}+\frac{\partial{{}^{2}}{\hat{g}}}{\partial{\rho}^{2}}\right)+(\alpha-2)\frac{\partial{\hat{g}}}{\partial{\rho}}=0.$

Consider a special class of solutions of Eq. $(\ref{Euler-Poisson-Darboux
equation-alpha})$ under condition of separation of variables
$g(x_{0},\rho)=\Xi(x_{0})\Upsilon(\rho)$:

$\frac{1}{\Xi}\frac{d{{}^{2}}{\Xi}}{d{x_{0}}^{2}}+\frac{1}{\Upsilon}\frac{d{{}^{2}}{\Upsilon}}{d{\rho}^{2}}-\frac{(\alpha-2)}{\Upsilon\rho}\frac{d{\Upsilon}}{d{\rho}}=0.$

Relations

$\displaystyle-\frac{1}{\Xi}\frac{d{{}^{2}}{\Xi}}{d{x_{0}}^{2}}=\frac{1}{\Upsilon}\frac{d{{}^{2}}{\Upsilon}}{d{\rho}^{2}}-\frac{(\alpha-2)}{\Upsilon\rho}\frac{d{\Upsilon}}{d{\rho}}=-\breve{\beta}^{2}\
\ \ \ \ (\breve{\beta}=const\in\mathbf{R})$

are equivalent to the following system of ordinary differential equations:

$\left\\{\begin{array}[]{l}\frac{d{{}^{2}}{\Xi}}{d{x_{0}}^{2}}-\breve{\beta}^{2}\Xi=0,\\\
\rho^{2}\frac{d{{}^{2}}{\Upsilon}}{d{\rho}^{2}}-(\alpha-2)\rho\frac{d{\Upsilon}}{d{\rho}}+\breve{\beta}^{2}\rho^{2}\Upsilon=0.\end{array}\right.$
(2.11)

The first equation of the system $(\ref{eq-sep-x_2-x_0-hyper-cyl})$ may be
solved using hyperbolic functions: $\
\Xi_{\breve{\beta}}(x_{0})=b^{1}_{\breve{\beta}}\cosh{\breve{\beta}x_{0}}+b^{2}_{\breve{\beta}}\sinh{\breve{\beta}x_{0}}$;
$\ b^{1}_{\breve{\beta}},b^{2}_{\breve{\beta}}=const\in\mathbf{R}$. In
particular, values $b^{1}_{\breve{\beta}}=b^{2}_{\breve{\beta}}=1$ imply that
$\ \Xi_{\breve{\beta}}(x_{0})=e^{\breve{\beta}x_{0}}$ (see, e.g., [9]).

The second equation of the system $(\ref{eq-sep-x_2-x_0-hyper-cyl})$ may be
solved using linear independent solutions

$\Upsilon_{\breve{\beta}}(\rho)={\rho}^{\frac{\alpha-1}{2}}\left[a^{1}_{\breve{\beta}}J_{\frac{\alpha-1}{2}}(\breve{\beta}\rho)+a^{2}_{\breve{\beta}}Y_{\frac{\alpha-1}{2}}(\breve{\beta}\rho)\right]$;
$\ a^{1}_{\breve{\beta}}$, $a^{2}_{\breve{\beta}}=const\in\mathbf{R}$,
where $J_{\frac{\alpha-1}{2}}(\breve{\beta}\rho)$ and
$Y_{\frac{\alpha-1}{2}}(\breve{\beta}\rho)$ are Bessel functions of the first
and second kind of order $\frac{\alpha-1}{2}$ and real argument
$\breve{\beta}\rho$ (see, e.g., [61, 21, 53]).

##  3\. Potential Meridional Fields in $\mathbb{R}^{4}$ and Four-Dimensional
$\alpha$-Meridional Mappings of the Second Kind

Equation (2.9) in cylindrical coordinates within potential meridional fields
in $\mathbb{R}^{4}$ leads to a family of Vekua type systems for different
values of the parameter $\alpha$ investigated by Sommen et al. in the context
of the theory of _Quaternion-valued monogenic functions of axial type_ (see,
e.g., [49, 50, 23])

$\displaystyle\begin{cases}\rho\left(\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{\rho}}}{\partial{\rho}}\right)+(\alpha-2)u_{\rho}=0,\\\\[4.30554pt]
\frac{\partial{u_{0}}}{\partial{\rho}}=-\frac{\partial{u_{\rho}}}{\partial{x_{0}}}.\end{cases}$
(3.1)

We should take into account that in our setting
$u_{0}=\frac{\partial{g}}{\partial{x_{0}}},\
u_{\rho}=-\frac{\partial{g}}{\partial{\rho}}.$

The static system (2.6) is reduced to the following two-dimensional system in
the meridian half-plane:

$\displaystyle\begin{cases}\rho\left(\frac{\partial{V_{0}}}{\partial{x_{0}}}+\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)-(\alpha-2)V_{\rho}=0,\\\\[4.30554pt]
\frac{\partial{V_{0}}}{\partial{\rho}}=\frac{\partial{V_{\rho}}}{\partial{x_{0}}},\end{cases}$
(3.2)

where $V_{0}=u_{0}$, $V_{\rho}=-u_{\rho}$, and
$V_{1}=V_{\rho}\frac{x_{1}}{\rho}$, $V_{2}=V_{\rho}\frac{x_{2}}{\rho}$,
$V_{3}=V_{\rho}\frac{x_{3}}{\rho}$.

The principal invariants of the Jacobian matrix $\mathbf{J}(\vec{V}(x))$ may
be demonstrated explicitly. It should be noted that the Jacobian matrix (1.14)
is substantially simplified:

$\displaystyle\tiny{\begin{pmatrix}\left[-\frac{\partial{V_{\rho}}}{\partial{\rho}}+\frac{V_{\rho}}{\rho}(\alpha-2)\right]&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{1}}{\rho}&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{2}}{\rho}&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{3}}{\rho}\\\\[2.15277pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{1}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{1}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{2}^{2}+x_{3}^{2}}{\rho^{2}}\right)&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{2}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{3}}{\rho^{2}}\\\\[2.15277pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{2}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{2}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{2}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{1}^{2}+x_{3}^{2}}{\rho^{2}}\right)&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{2}x_{3}}{\rho^{2}}\\\\[2.15277pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{3}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{3}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{2}x_{3}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{3}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{1}^{2}+x_{2}^{2}}{\rho^{2}}\right)\end{pmatrix}}$
(3.3)

######  Theorem 3.1.

Roots of the characteristic equation (1.15) of the Jacobian matrix (3.3) are
given by exact formulas:

$\displaystyle\lambda_{0,1}$ $\displaystyle=\frac{V_{\rho}}{\rho},$
$\displaystyle\lambda_{2,3}$
$\displaystyle=\frac{(\alpha-2)}{2}\frac{V_{\rho}}{\rho}\pm$
$\displaystyle\hskip
21.52771pt\sqrt{\frac{(\alpha-2)^{2}}{4}\left(\frac{V_{\rho}}{\rho}\right)^{2}-(\alpha-2)\frac{V_{\rho}}{\rho}\frac{\partial{V_{\rho}}}{\partial{\rho}}+\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}}.$

###### Proof.

The principal invariants of the Jacobian matrix (3.3) are written as

$\displaystyle I_{\mathbf{J}(\vec{V})}$
$\displaystyle=\alpha\frac{V_{\rho}}{\rho},$ $\displaystyle
II_{\mathbf{J}(\vec{V})}$
$\displaystyle=-\left[\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}\right]+(\alpha-2)\frac{V_{\rho}}{\rho}\frac{\partial{V_{\rho}}}{\partial{\rho}}+(2\alpha-3)\left(\frac{V_{\rho}}{\rho}\right)^{2},$
$\displaystyle III_{\mathbf{J}(\vec{V})}$
$\displaystyle=-2\frac{V_{\rho}}{\rho}\left[\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}\right]+(\alpha-2)\left(\frac{V_{\rho}}{\rho}\right)^{2}\left(2\frac{\partial{V_{\rho}}}{\partial{\rho}}+\frac{V_{\rho}}{\rho}\right),$
$\displaystyle IV_{\mathbf{J}(\vec{V})}$
$\displaystyle=-\left(\frac{V_{\rho}}{\rho}\right)^{2}\left[\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}\right]+(\alpha-2)\left(\frac{V_{\rho}}{\rho}\right)^{3}\frac{\partial{V_{\rho}}}{\partial{\rho}}.$

The characteristic equation (1.15) into the framework of the system (3.2) may
be factored:

$\displaystyle\left(\lambda-\frac{V_{\rho}}{\rho}\right)^{2}\times$
$\displaystyle\left[\lambda^{2}-(\alpha-2)\frac{V_{\rho}}{\rho}\lambda+(\alpha-2)\frac{V_{\rho}}{\rho}\frac{\partial{V_{\rho}}}{\partial{\rho}}-\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}-\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}\right]=0.\qed$

######  Corollary 3.2 (On the set of degenerate points).

Assume that a potential meridional field
$\vec{V}=(V_{0},\frac{x_{1}}{\rho}V_{\rho},\frac{x_{2}}{\rho}V_{\rho},\frac{x_{3}}{\rho}V_{\rho})$
satisfies the system (3.2). The set of degenerate points of the Jacobian
matrix (3.3) is provided by two independent equations:

${V_{\rho}}=0,\quad\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}-(\alpha-2)\frac{V_{\rho}}{\rho}\frac{\partial{V_{\rho}}}{\partial{\rho}}=0.$

######  Corollary 3.3 (On the zero divergence condition).

Assume that a potential meridional field
$\vec{V}=(V_{0},\frac{x_{1}}{\rho}V_{\rho},\frac{x_{2}}{\rho}V_{\rho},\frac{x_{3}}{\rho}V_{\rho})$
satisfies the system (3.2), where $\alpha\neq 0$. Every point
$x=(x_{0},x_{1},x_{2},x_{3})$, where
$\mathrm{div}\,\vec{V}(x_{0},x_{1},x_{2},x_{3})=0$, is a degenerate point of
the Jacobian matrix (3.3).

######  Remark 3.4.

The second formula of Theorem 3.1 may be simplified:

$\displaystyle\lambda_{2,3}$
$\displaystyle=\frac{(\alpha-2)}{2}\frac{V_{\rho}}{\rho}\pm\sqrt{\left(\frac{\alpha-2}{2}\frac{V_{\rho}}{\rho}-\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}}.$

Geometric properties of the the Jacobian matrix (3.3) allow us to introduce
the concept of four-dimensional $\alpha$-meridional mappings of the first and
second kind.

######  Definition 3.5.

Let $\alpha$ be a real parameter, while $\Lambda\subset\mathbb{R}^{4}$ be a
simply connected open domain, where $x_{1}\neq 0,x_{2}\neq 0,x_{3}\neq 0$.
Assume that an exact solution $(u_{0},u_{1},u_{2},u_{3})$ of the system (2.7),
where $\alpha\neq 0$, satisfies axially symmetric conditions
${u_{1}}{x_{2}}={u_{2}}{x_{1}},\quad{u_{1}}{x_{3}}={u_{3}}{x_{1}},\quad{u_{2}}{x_{3}}={u_{3}}{x_{2}}$
in $\Lambda$. Then mapping
$u=u_{0}+iu_{1}+ju_{2}+ku_{3}:\Lambda\rightarrow\mathbb{R}^{4}$ is called
four-dimensional $\alpha$-meridional mapping of the first kind, while mapping
$\overline{u}=u_{0}-iu_{1}-ju_{2}-ku_{3}:\Lambda\rightarrow\mathbb{R}^{4}$ is
called four-dimensional $\alpha$-meridional mapping of the second kind.

The set of degenerate points of every four-dimensional $\alpha$-meridional
mapping of the second kind coincides with the set of degenerate points of the
Jacobian matrix (3.3).

######  Remark 3.6.

The concept of four-dimensional $\alpha$-meridional mappings of the first and
second kind may be efficiently developed in the context of the theory of
_Hyperbolic function theory in the skew-field of quaternions_ and the theory
of _Modified harmonic functions in $\mathbb{R}^{4}$_ under conditions of
$\frac{\partial{h}}{\partial{\theta}}=0$,
$\frac{\partial{h}}{\partial{\psi}}=0$.

Let us first look at the basic properties of potential meridional fields in
the context of the theory of _Modified harmonic functions in $\mathbb{R}^{4}$_
in case $\alpha=-2$, where the systems (3.1), (3.2) are expressed as

$\displaystyle\begin{cases}\rho\left(\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{\rho}}}{\partial{\rho}}\right)-4u_{\rho}=0,\\\
\frac{\partial{u_{0}}}{\partial{\rho}}=-\frac{\partial{u_{\rho}}}{\partial{x_{0}}},\end{cases}$
$\displaystyle\begin{cases}\rho\left(\frac{\partial{V_{0}}}{\partial{x_{0}}}+\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)+4V_{\rho}=0,\\\
\frac{\partial{V_{0}}}{\partial{\rho}}=\frac{\partial{V_{\rho}}}{\partial{x_{0}}},\end{cases}$
$\displaystyle\rho\left(\frac{\partial{{}^{2}}{g}}{\partial{x_{0}}^{2}}+\frac{\partial{{}^{2}}{g}}{\partial{\rho}^{2}}\right)+4\frac{\partial{g}}{\partial{\rho}}=0,\quad\quad\rho\left(\frac{\partial{{}^{2}}{\hat{g}}}{\partial{x_{0}}^{2}}+\frac{\partial{{}^{2}}{\hat{g}}}{\partial{\rho}^{2}}\right)-4\frac{\partial{\hat{g}}}{\partial{\rho}}=0.$

The Jacobian matrix (3.3) takes the following form:

$\displaystyle\small{\begin{pmatrix}\left[-\frac{\partial{V_{\rho}}}{\partial{\rho}}-4\frac{V_{\rho}}{\rho}\right]&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{1}}{\rho}&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{2}}{\rho}&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{3}}{\rho}\\\\[3.87498pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{1}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{1}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{2}^{2}+x_{3}^{2}}{\rho^{2}}\right)&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{2}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{3}}{\rho^{2}}\\\\[3.87498pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{2}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{2}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{2}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{1}^{2}+x_{3}^{2}}{\rho^{2}}\right)&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{2}x_{3}}{\rho^{2}}\\\\[3.87498pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{3}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{3}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{2}x_{3}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{3}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{1}^{2}+x_{2}^{2}}{\rho^{2}}\right)\end{pmatrix}}$
(3.4)

######  Corollary 3.7.

Roots of the characteristic equation (1.15) are given by the formulas

$\displaystyle\lambda_{0,1}=\frac{V_{\rho}}{\rho},\;\lambda_{2,3}=-2\frac{V_{\rho}}{\rho}\pm\sqrt{\left(2\frac{V_{\rho}}{\rho}+\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}}.$
(3.5)

Exact formulas (3.5) demonstrate explicitly the geometric specifics of the
Jacobian matrix (3.4).

######  Corollary 3.8.

Suppose that $\alpha=-2$,
$\frac{V_{\rho}}{\rho}=\frac{V_{1}}{x_{1}}=\frac{V_{2}}{x_{2}}=\frac{V_{3}}{x_{3}}$.
The set of degenerate points of the Jacobian matrix (3.4) is provided by two
independent equations:

${V_{\rho}}=0,\quad\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}+4\frac{V_{\rho}}{\rho}\frac{\partial{V_{\rho}}}{\partial{\rho}}=0.$

######  Example.

Consider a generalized axially symmetric potential in case $\alpha=-2$ using
Bessel function of the first kind of order $-\frac{3}{2}$ and real argument
$\breve{\beta}\rho$:

$g(x_{0},\rho)=e^{\breve{\beta}x_{0}}\rho^{-\frac{3}{2}}J_{-\frac{3}{2}}(\breve{\beta}\rho),$

where $\rho>0$. We deal with analytic models of 2-modified harmonic meridional
fields
$\vec{V}=(V_{0},\frac{x_{1}}{\rho}V_{\rho},\frac{x_{2}}{\rho}V_{\rho},\frac{x_{3}}{\rho}V_{\rho})$,
where

$\displaystyle
V_{0}=\breve{\beta}e^{\breve{\beta}x_{0}}\rho^{-\frac{3}{2}}J_{-\frac{3}{2}}(\breve{\beta}\rho),\quad
V_{\rho}=e^{\breve{\beta}x_{0}}\rho^{-\frac{3}{2}}\left(J^{\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)-\frac{3}{2\rho}J_{-\frac{3}{2}}(\breve{\beta}\rho)\right),$

such that

$\displaystyle\frac{\partial{V_{\rho}}}{\partial{x_{0}}}$
$\displaystyle=\breve{\beta}e^{\breve{\beta}x_{0}}\rho^{-\frac{3}{2}}\left(J^{\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)-\frac{3}{2\rho}J_{-\frac{3}{2}}(\breve{\beta}\rho)\right),$
$\displaystyle\frac{\partial{V_{\rho}}}{\partial{\rho}}$
$\displaystyle=e^{\breve{\beta}x_{0}}\rho^{-\frac{3}{2}}\left(J^{\prime\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)-\frac{3}{\rho}J^{\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)+\frac{15}{4\rho^{2}}J_{-\frac{3}{2}}(\breve{\beta}\rho)\right).$

Roots of the characteristic equation (1.15) of the Jacobian matrix (3.4) are
given by the formulas

$\displaystyle\lambda_{0,1}=\frac{e^{\breve{\beta}x_{0}}}{\rho^{2}\sqrt{\rho}}\left(J^{\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)-\frac{3}{2\rho}J_{-\frac{3}{2}}(\breve{\beta}\rho)\right),$
$\displaystyle\lambda_{2,3}=-2\frac{e^{\breve{\beta}x_{0}}}{\rho^{2}\sqrt{\rho}}\left(J^{\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)-\frac{3}{2\rho}J_{-\frac{3}{2}}(\breve{\beta}\rho)\right)\pm$
$\displaystyle\frac{e^{\breve{\beta}x_{0}}}{\rho\sqrt{\rho}}\sqrt{\breve{\beta}^{2}\left(J^{\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)-\frac{3}{2\rho}J_{-\frac{3}{2}}(\breve{\beta}\rho)\right)^{2}+\left(J^{\prime\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)-\frac{1}{\rho}J^{\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)+\frac{3}{4\rho^{2}}J_{-\frac{3}{2}}(\breve{\beta}\rho)\right)^{2}}.$

The set of degenerate points of the Jacobian matrix (3.4) is provided by two
independent equations:

$\displaystyle
J^{\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)-\frac{3}{2\rho}J_{-\frac{3}{2}}(\breve{\beta}\rho)=0,$
$\displaystyle\left(\breve{\beta}^{2}-\frac{4}{\rho^{2}}\right)\left(J^{\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)-\frac{3}{2\rho}J_{-\frac{3}{2}}(\breve{\beta}\rho)\right)^{2}+\left(J^{\prime\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)-\frac{1}{\rho}J^{\prime}_{-\frac{3}{2}}(\breve{\beta}\rho)+\frac{3}{4\rho^{2}}J_{-\frac{3}{2}}(\breve{\beta}\rho)\right)^{2}=0.$

##  4\. The Radially Holomorphic Potential in $\mathbb{R}^{4}$ and Geometric
Properties of Quaternionic Möbius Transformations with Real Coefficients

When $\alpha=2$, the system (1.5) in cylindrical coordinates within Fueter’s
construction in $\mathbb{R}^{4}$ (1.4) is reduced to the Cauchy-Riemann type
system in the meridian half-plane (see, e.g., [2, 32, 13, 16, 9, 10])

$\displaystyle\begin{cases}\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{\rho}}}{\partial{\rho}}=0,\\\\[4.30554pt]
\frac{\partial{u_{0}}}{\partial{\rho}}=-\frac{\partial{u_{\rho}}}{\partial{x_{0}}},\end{cases}$

where $u_{0}=\frac{\partial{g}}{\partial{x_{0}}},\
u_{\rho}=-\frac{\partial{g}}{\partial{\rho}}$. The system (1.7) is expressed
as

$\displaystyle\begin{cases}\frac{\partial{V_{0}}}{\partial{x_{0}}}+\frac{\partial{V_{\rho}}}{\partial{\rho}}=0,\\\
\frac{\partial{V_{0}}}{\partial{\rho}}=\frac{\partial{V_{\rho}}}{\partial{x_{0}}}.\end{cases}$
(4.1)

The generalized Stokes-Beltrami system in the meridian half-plane (2.10)
becomes the Cauchy-Riemann type system in the meridian half-plane concerning
functions $g=g(x_{0},\rho)$, $\hat{g}=\hat{g}(x_{0},\rho)$:

$\displaystyle\begin{cases}\frac{\partial{g}}{\partial{x_{0}}}-\frac{\partial{\hat{g}}}{\partial{\rho}}=0,\\\\[4.30554pt]
\frac{\partial{g}}{\partial{\rho}}=-\frac{\partial{\hat{g}}}{\partial{x_{0}}}.\end{cases}$
(4.2)

Generalized axially symmetric potential $g=g(x_{0},\rho)$ and the Stokes
stream function $\hat{g}=\hat{g}(x_{0},\rho)$ satisfy equations

$\displaystyle\frac{\partial{{}^{2}}{g}}{\partial{x_{0}}^{2}}+\frac{\partial{{}^{2}}{g}}{\partial{\rho}^{2}}=0,\quad\frac{\partial{{}^{2}}{\hat{g}}}{\partial{x_{0}}^{2}}+\frac{\partial{{}^{2}}{\hat{g}}}{\partial{\rho}^{2}}=0.$

On the other hand, an important concept of radially holomorphic functions in
$\mathbb{R}^{4}$ was introduced by Gürlebeck, Habetha, Sprößig in 2008 in the
context of the theory of _Holomorphic functions in n-dimensional space_ [32].

######  Definition 4.1.

The radial differential operator in $\mathbb{R}^{4}$ is defined by formula

$\partial_{rad}{G}:=\frac{1}{2}\left(\frac{\partial{}}{\partial{x_{0}}}-I\frac{\partial{}}{\partial{\rho}}\right)G:=G^{\prime}\
\ \ \ \ (G=g+I\hat{g}).$

Every quaternion-valued function $G=g+I\hat{g}$ satisfying a Cauchy-Riemann
type differential equation in $\Lambda$ $(\rho>0)$

$\displaystyle\overline{\partial}_{rad}{G}:=\frac{1}{2}\left(\frac{\partial{}}{\partial{x_{0}}}+I\frac{\partial{}}{\partial{\rho}}\right)G=0$
(4.3)

is called a radially holomorphic in $\Lambda$, while quaternion-conjugate
function $\overline{G}=g-I\hat{g}$ is called a radially anti-holomorphic in
$\Lambda$.

It is easy to see that general class of exact solutions of Eq. (4.3) is
equivalently represented as general class of exact solutions of the Cauchy-
Riemann type system in the meridian half-plane concerning functions
$g=g(x_{0},\rho)$, $\hat{g}=\hat{g}(x_{0},\rho)$ (4.2) (see, e.g., [10]).

The notation $\partial_{rad}{G}:=G^{\prime}$ has been justified in [32] by
some clear statements. In particular, Eq. (4.3) implies that

$\displaystyle G^{\prime}=\frac{\partial{G}}{\partial{x_{0}}}.$ (4.4)

######  Definition 4.2.

Suppose that a radially holomorphic function $G=g+I\hat{g}$ in $\Lambda$
satisfies a differential equation

$\displaystyle G^{\prime}=F,$

where $F=u_{0}+Iu_{\rho}$ characterizes a radially holomorphic function in
$\Lambda$. The function $G$ is called a radially holomorphic primitive of $F$
in $\Lambda$.

Elementary radially holomorphic functions in $\mathbb{R}^{4}$ may be
introduced as elementary functions of a quaternionic variable
$G=G(x)=g(x_{0},\rho)+I\hat{g}_{\rho}(x_{0},\rho)$ satisfying the following
relations:

$\displaystyle[x^{n}$
$\displaystyle:=r^{n}(\cos{n\varphi}+I\sin{n\varphi})]^{\prime}=nx^{n-1};$
$\displaystyle[e^{x}$
$\displaystyle:=e^{x_{0}}(\cos{\rho}+I\sin{\rho})]^{\prime}=e^{x};$
$\displaystyle[\cos{x}$
$\displaystyle:=\frac{1}{2}(e^{-Ix}+e^{Ix})]^{\prime}=-\sin{x};$
$\displaystyle[\sin{x}$
$\displaystyle:=\frac{I}{2}(e^{-Ix}-e^{Ix})]^{\prime}=\cos{x};$
$\displaystyle[\ln{x}$ $\displaystyle:=\ln r+I\varphi]^{\prime}=x^{-1}.$

Applications of radially holomorphic functions in $\mathbb{R}^{4}$ have been
missed in the next book of Gürlebeck, Habetha, Sprößig [33]. Tools of the
radially holomorphic potential in $\mathbb{R}^{4}$ in conjunction with tools
of the theory of _Static potential meridional vector fields in
$\mathbb{R}^{4}$_ allow us to make up for the gap.

######  Definition 4.3.

Radially holomorphic primitive $G=g+I\hat{g}$ in simply connected open domains
$\Lambda$ $(\rho>0)$ into the framework of the systems (4.1), (4.2) is called
the radially holomorphic potential in $\mathbb{R}^{4}$.

When $\alpha=2$, the principal invariants of the Jacobian matrix (3.3)

$\displaystyle\begin{pmatrix}-\frac{\partial{V_{\rho}}}{\partial{\rho}}&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{1}}{\rho}&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{2}}{\rho}&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{3}}{\rho}\\\\[4.30554pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{1}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{1}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{2}^{2}+x_{3}^{2}}{\rho^{2}}\right)&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{2}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{3}}{\rho^{2}}\\\\[4.30554pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{2}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{2}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{2}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{1}^{2}+x_{3}^{2}}{\rho^{2}}\right)&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{2}x_{3}}{\rho^{2}}\\\\[4.30554pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{3}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{3}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{2}x_{3}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{3}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{1}^{2}+x_{2}^{2}}{\rho^{2}}\right)\end{pmatrix}$
(4.5)

are written as

$\displaystyle I_{\mathbf{J}(\vec{V})}=2\frac{V_{\rho}}{\rho},\quad
II_{\mathbf{J}(\vec{V})}=-\left[\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}\right]+\left(\frac{V_{\rho}}{\rho}\right)^{2},$
$\displaystyle
III_{\mathbf{J}(\vec{V})}=-2\frac{V_{\rho}}{\rho}\left[\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}\right],$
$\displaystyle
IV_{\mathbf{J}(\vec{V})}=-\left(\frac{V_{\rho}}{\rho}\right)^{2}\left[\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}\right].$

######  Corollary 4.4.

Roots of the characteristic equation (1.15) of the Jacobian matrix (4.5) are
given by the formulas

$\displaystyle\lambda_{0,1}=\frac{V_{\rho}}{\rho},\ \ \ \ \
\lambda_{2,3}=\pm\sqrt{\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}}=\pm|F^{\prime}|.$
(4.6)

Exact formulas (4.6) demonstrate explicitly the geometric specifics of the
Jacobian matrix (4.5).

Remarkable extensions of the basic concepts of Möbius transformations in four
dimensions (see, e.g., [1, 41]) have been developed, in particular, by Porter
[54], Cao [11], Parker and Short [48], Bisi and Gentili [4], Stoppato [57].

Meanwhile, geometric properties of quaternionic Möbius transformations with
real coefficients $a,b,c,d$ within Fueter’s construction in $\mathbb{R}^{4}$
(1.4) $F(x)=(ax+b)(cx+d)^{-1}$, where $ad-bc=1$, may be characterized in the
context of the theory of _Static potential meridional vector fields in
$\mathbb{R}^{4}$_.

######  Example.

Consider quaternionic Möbius transformations with real coefficients, where
$c=1$, $ad-b=1$. Radially anti-holomorphic function is expressed as
$\overline{F}(x)=-(\overline{x}+d)^{-1}+a$. The radially holomorphic potential
in $\mathbb{R}^{4}$ is written as

$G=-\ln{(x+d)}+ax.$

We deal with analytic models of meridional fields
$\vec{V}=(V_{0},\frac{x_{1}}{\rho}V_{\rho},\frac{x_{2}}{\rho}V_{\rho},\frac{x_{3}}{\rho}V_{\rho})$,
where

$\displaystyle V_{0}=-\frac{x_{0}+d}{(x_{0}+d)^{2}+\rho^{2}}+a,\quad
V_{\rho}=-\frac{\rho}{(x_{0}+d)^{2}+\rho^{2}},$
$\displaystyle\frac{\partial{V_{\rho}}}{\partial{x_{0}}}=\frac{2\rho(x_{0}+d)}{[(x_{0}+d)^{2}+\rho^{2}]^{2}},\quad\frac{\partial{V_{\rho}}}{\partial{\rho}}=\frac{-(x_{0}+d)^{2}+\rho^{2}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}.$

The Jacobian matrix (4.5) takes the following form:

$\displaystyle\begin{pmatrix}\frac{(x_{0}+d)^{2}-\rho^{2}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{2(x_{0}+d)x_{1}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{2(x_{0}+d)x_{2}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{2(x_{0}+d)x_{3}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}\\\\[4.30554pt]
\frac{2(x_{0}+d)x_{1}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{-(x_{0}+d)^{2}-\rho^{2}+2x_{1}^{2}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{2x_{1}x_{2}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{2x_{1}x_{3}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}\\\\[4.30554pt]
\frac{2(x_{0}+d)x_{2}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{2x_{1}x_{2}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{-(x_{0}+d)^{2}-\rho^{2}+2x_{2}^{2}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{2x_{2}x_{3}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}\\\\[4.30554pt]
\frac{2(x_{0}+d)x_{3}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{2x_{1}x_{3}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{2x_{2}x_{3}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}&\frac{-(x_{0}+d)^{2}-\rho^{2}+2x_{3}^{2}}{[(x_{0}+d)^{2}+\rho^{2}]^{2}}\end{pmatrix}$
(4.7)

Roots of the characteristic equation (1.15) of the Jacobian matrix (4.7) are
given by the formulas

$\displaystyle\lambda_{0,1}=-\frac{1}{(x_{0}+d)^{2}+\rho^{2}},\quad\lambda_{2,3}=\pm\frac{1}{(x_{0}+d)^{2}+\rho^{2}}.$

Thus, the set of degenerate points of the Jacobian matrix (4.7) is empty.

Geometric properties of elementary radially holomorphic functions in
$\mathbb{R}^{4}$ raise new issues for consideration.

##  5\. Analytic Models of Potential Meridional Fields in $\mathbb{R}^{4}$
Generated by Bessel Functions of the First Kind of Integer Order and
Quaternionic Argument

First outlines of the concept of integral transforms with quaternion-valued
kernels within Fueter’s construction in $\mathbb{R}^{4}$ were presented by the
author in 2003 [8]. Now tools of the Fourier-Fueter cosine and sine transforms
of real-valued original functions $\tilde{\eta}=\tilde{\eta}(\tau)$ allow us
to obtain integral representations for Bessel functions of the first kind of
integer order $n$ and quaternionic argument $x$.

In accordance with the basic concepts of the theory of functions of a complex
variable and the one-sided Laplace transform (see, e.g., [22, 40, 7]), every
complex-valued original function $\tilde{\eta}=\tilde{\eta}(\tau)$ of a real
argument $\tau$ satisfies the following conditions:

  1. (1)

the function $\tilde{\eta}(\tau)$ satisfies the Hölder’s condition for any
$\tau$ except for points $\tau=\tau_{1},\tau_{2},\ldots$, where the function
$\tilde{\eta}(\tau)$ has discontinuities of the first kind (there exists a
finite number of such points for every finite interval),

  2. (2)

the function $\tilde{\eta}(\tau)=0$ for any $\tau<0$,

  3. (3)

$|\tilde{\eta}(\tau)|<Me^{s_{0}\tau}$ with certain constants $M>0,s_{0}\geq 0$
for any $\tau\geq 0$.

In accordance with the Hölder’s condition, there exist certain constants
$A>0$, $0<\gamma\leq 1$, $\delta_{0}>0$ such that
$|\tilde{\eta}(\tau+\delta)-\tilde{\eta}(\tau)|\leq A|\delta|^{\gamma}$ for
any $\delta$, where $|\delta|\leq\delta_{0}$. The number $s_{0}$ is referred
to as the growth rate of the function $\tilde{\eta}(\tau)$; $s_{0}$ may take
the value 0 in case of bounded functions $\tilde{\eta}(\tau)$.

To extend some concepts of integral transforms of real-valued original
functions $\tilde{\eta}=\tilde{\eta}(\tau)$ within Fueter’s construction in
$\mathbb{R}^{4}$ (1.4), a correspondence between two functions
$\tilde{\eta}(\tau)$ and $F(x)$ in the form

$\displaystyle F(x):=\int_{0}^{\infty}\tilde{\eta}(\tau)K(x,\tau)d\tau,$

where quaternion-valued function $K(x,\tau)$ belongs to Fueter’s construction
in $\mathbb{R}^{4}$ for any $\tau\geq 0$, shall henceforth be referred to as
integral transform with quaternion-valued kernel $K(x,\tau)$.

######  Definition 5.1.

Let $\tilde{\eta}(\tau)$ be a real-valued original function and $\rho>0$.
Every integral transform with quaternion-valued kernel $K(x,\tau)=e^{-x\tau}$

$\displaystyle
F(x):=\mathfrak{LF}\\{\tilde{\eta}(\tau);x\\}=\int_{0}^{\infty}\tilde{\eta}(\tau)e^{-x\tau}d\tau$

is called the one-sided Laplace-Fueter transform of $\tilde{\eta}(\tau)$.

Analytic properties of radially anti-holomorphic functions in $\mathbb{R}^{4}$

$\displaystyle\overline{F}(x)=\int_{0}^{\infty}\tilde{\eta}(\tau)\overline{e^{-x\tau}}d\tau=\int_{0}^{\infty}\tilde{\eta}(\tau)e^{-x_{0}\tau}[\cos(\rho\tau)+I\sin(\rho\tau)]d\tau$

are of particular interest to the theory of _Potential meridional vector
fields in $\mathbb{R}^{4}$_.

######  Definition 5.2.

Let $\tilde{\eta}(\tau)$ be a real-valued original function and $\rho>0$.
Every integral transform with quaternion-valued kernel $K(x,\tau)=\cos(x\tau)$

$\displaystyle
F(x):=\mathfrak{FF}c\\{\tilde{\eta}(\tau);x\\}=\int_{0}^{\infty}\tilde{\eta}(\tau)\cos(x\tau)d\tau=\frac{1}{2}\int_{0}^{\infty}\tilde{\eta}(\tau)(e^{-Ix\tau}+e^{Ix\tau})d\tau$

is called the Fourier-Fueter cosine transform of $\tilde{\eta}(\tau)$.

######  Remark 5.3.

Consider an independent quaternionic variable of the following form:
$y=Ix=-\rho+Ix_{0}$. The Fourier-Fueter cosine transform of
$\tilde{\eta}(\tau)$ may be equivalently represented by means of the one-sided
Laplace-Fueter transform of $\tilde{\eta}(\tau)$:

$\displaystyle\mathfrak{FF}c\\{\tilde{\eta}(\tau);x\\}=\frac{1}{2}[\mathfrak{LF}\\{\tilde{\eta}(\tau);y\\}+\mathfrak{LF}\\{\tilde{\eta}(\tau);-y\\}].$

Analytic models of potential meridional fields generated by the Fourier-Fueter
cosine transform of $\tilde{\eta}(\tau)$ are described as
$\vec{V}=(V_{0},\frac{x_{1}}{\rho}V_{\rho},\frac{x_{2}}{\rho}V_{\rho},\frac{x_{3}}{\rho}V_{\rho})$,
where

$\displaystyle
V_{0}=\int_{0}^{\infty}\tilde{\eta}(\tau)\cosh(\rho\tau)\cos(x_{0}\tau)d\tau,\quad
V_{\rho}=\int_{0}^{\infty}\tilde{\eta}(\tau)\sinh(\rho\tau)\sin(x_{0}\tau)d\tau.$

######  Definition 5.4.

Let $\tilde{\eta}(\tau)$ be a real-valued original function and $\rho>0$.
Every integral transform with quaternion-valued kernel $K(x,\tau)=\sin(x\tau)$

$\displaystyle
F(x):=\mathfrak{FF}s\\{\tilde{\eta}(\tau);x\\}=\int_{0}^{\infty}\tilde{\eta}(\tau)\sin(x\tau)d\tau=\frac{I}{2}\int_{0}^{\infty}\tilde{\eta}(\tau)(e^{-Ix\tau}-e^{Ix\tau})d\tau$

is called the Fourier-Fueter sine transform of $\tilde{\eta}(\tau)$.

######  Remark 5.5.

The Fourier-Fueter sine transform of $\tilde{\eta}(\tau)$ may be equivalently
represented by means of the one-sided Laplace-Fueter transform of
$\tilde{\eta}(\tau)$:

$\displaystyle\mathfrak{FF}s\\{\tilde{\eta}(\tau);x\\}=\frac{I}{2}[\mathfrak{LF}\\{\tilde{\eta}(\tau);y\\}-\mathfrak{LF}\\{\tilde{\eta}(\tau);-y\\}].$

Analytic models of potential meridional fields generated by the Fourier-Fueter
sine transform of $\tilde{\eta}(\tau)$ are described as
$\vec{V}=(V_{0},\frac{x_{1}}{\rho}V_{\rho},\frac{x_{2}}{\rho}V_{\rho},\frac{x_{3}}{\rho}V_{\rho})$,
where

$\displaystyle
V_{0}=\int_{0}^{\infty}\tilde{\eta}(\tau)\cosh(\rho\tau)\sin(x_{0}\tau)d\tau,\quad
V_{\rho}=-\int_{0}^{\infty}\tilde{\eta}(\tau)\sinh(\rho\tau)\cos(x_{0}\tau)d\tau.$

Important properties of quaternionic power series with real coefficients
studied by Leutwiler and Hempfling in the context of _Modified quaternionic
analysis in $\mathbb{R}^{4}$_ (see, e.g., [41, 34]) allow us to extend every
Bessel function of the first kind of integer order and complex argument (see,
e.g., [61, 21, 22]) within Fueter’s construction in $\mathbb{R}^{4}$ (1.4)
from the disk of radius $r$ in the complex plane
$D_{r}=\\{(x_{0},x_{1}):x_{0}^{2}+x_{1}^{2}<r^{2}\\}$ to the ball of radius
$r$ in $\mathbb{R}^{4}$
$B_{r}=\\{(x_{0},x_{1},x_{2},x_{3}):x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}<r^{2}\\}$.
As a corollary, every Bessel function of the first kind of integer order $n$
and quaternionic argument is represented by a quaternionic power series with
real coefficients:

$\displaystyle
J_{n}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!(n+m)!}\left(\frac{x}{2}\right)^{n+2m}.$

Furthermore, every elementary radially holomorphic function
$\left(\frac{x}{2}\right)^{m}$ may be expanded in a series of Bessel functions
of the first kind of integer order $(m+2n)$ and quaternionic argument:

$\displaystyle\left(\frac{x}{2}\right)^{m}=\sum_{n=0}^{\infty}\frac{(m+2n)(m+n-1)!}{n!}J_{m+2n}(x)\quad(m=1,2,3,\ldots)$

Using the Fourier-Fueter cosine transform of $\
\tilde{\eta}(\tau)=\frac{\cos(2n\arccos\tau)}{\sqrt{1-\tau^{2}}}$, integral
representations for Bessel functions of the first kind of even integer order
and quaternionic argument may be obtained:

$\displaystyle\frac{\pi}{2}(-1)^{n}J_{2n}(x)=\mathfrak{FF}c\\{\tilde{\eta}(\tau);x\\}=\int_{0}^{1}\frac{\cos(2n\arccos\tau)}{\sqrt{1-\tau^{2}}}\cos(x\tau)d\tau.$

Using the Fourier-Fueter sine transform of
$\tilde{\eta}(\tau)=\frac{\cos[(2n+1)\arccos\tau]}{\sqrt{1-\tau^{2}}}$,
integral representations for Bessel functions of the first kind of odd integer
order and quaternionic argument may be obtained:

$\displaystyle\frac{\pi}{2}(-1)^{n}J_{2n+1}(x)=\mathfrak{FF}s\\{\tilde{\eta}(\tau);x\\}=\int_{0}^{1}\frac{\cos[(2n+1)\arccos\tau]}{\sqrt{1-\tau^{2}}}\sin(x\tau)d\tau.$

######  Example.

Bessel function of the first kind of order zero and quaternionic argument is
represented as
$J_{0}(x)=\sum_{m=0}^{\infty}\frac{(-1)^{m}}{(m!)^{2}}\left(\frac{x}{2}\right)^{2m}.$

Integral representation of Bessel function of the first kind of order zero and
quaternionic argument is given by the formula

$\displaystyle
J_{0}(x)=\frac{2}{\pi}\mathfrak{FF}c\\{\tilde{\eta}(\tau);x\\}=\frac{2}{\pi}\int_{0}^{1}\frac{1}{\sqrt{1-\tau^{2}}}\cos(x\tau)d\tau.$

As seen, remarkable analytic model of a potential meridional field generated
by Bessel function of the first kind of order zero and quaternionic argument
is described as
$\vec{V}=(V_{0},\frac{x_{1}}{\rho}V_{\rho},\frac{x_{2}}{\rho}V_{\rho},\frac{x_{3}}{\rho}V_{\rho})$,
where

$\displaystyle
V_{0}=\int_{0}^{1}\frac{\cosh(\rho\tau)}{\sqrt{1-\tau^{2}}}\cos(x_{0}\tau)d\tau,\quad
V_{\rho}=\int_{0}^{1}\frac{\sinh(\rho\tau)}{\sqrt{1-\tau^{2}}}\sin(x_{0}\tau)d\tau.$

##  6\. Four-Dimensional Harmonic Meridional Mappings of the Second Kind and
Gradient Dynamical Systems with Harmonic Potential

Some remarkable properties of the Hessian matrix $\mathbf{H}(h(x))$ of
harmonic potentials $h=h(x_{0},x_{1},x_{2},x_{3})$ in the general four-
dimensional setting were described by Yanushauskas in 1980 by means of
homogeneous harmonic polynomials [63].

The geometric specifics of the Hessian matrix $\mathbf{H}(h(x))$ within
harmonic potential meridional fields $\vec{V}=\mathrm{grad}\ h$ may be
efficiently studied using the Jacobian matrix (3.3) in case $\alpha=0$:

$\displaystyle\tiny{\begin{pmatrix}\left(-\frac{\partial{V_{\rho}}}{\partial{\rho}}-2\frac{V_{\rho}}{\rho}\right)&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{1}}{\rho}&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{2}}{\rho}&\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{3}}{\rho}\\\\[2.15277pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{1}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{1}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{2}^{2}+x_{3}^{2}}{\rho^{2}}\right)&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{2}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{3}}{\rho^{2}}\\\\[2.15277pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{2}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{2}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{2}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{1}^{2}+x_{3}^{2}}{\rho^{2}}\right)&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{2}x_{3}}{\rho^{2}}\\\\[2.15277pt]
\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\frac{x_{3}}{\rho}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{1}x_{3}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}-\frac{V_{\rho}}{\rho}\right)\frac{x_{2}x_{3}}{\rho^{2}}&\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\frac{x_{3}^{2}}{\rho^{2}}+\frac{V_{\rho}}{\rho}\frac{x_{1}^{2}+x_{2}^{2}}{\rho^{2}}\right)\end{pmatrix}.}$
(6.1)

Let us look at the basic properties of harmonic potential meridional fields,
where the systems (3.1), (3.2) are expressed as

$\displaystyle\begin{cases}\rho\left(\frac{\partial{u_{0}}}{\partial{x_{0}}}-\frac{\partial{u_{\rho}}}{\partial{\rho}}\right)-2u_{\rho}=0,\\\
\frac{\partial{u_{0}}}{\partial{\rho}}=-\frac{\partial{u_{\rho}}}{\partial{x_{0}}},\end{cases}$
$\displaystyle\begin{cases}\rho\left(\frac{\partial{V_{0}}}{\partial{x_{0}}}+\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)+2V_{\rho}=0,\\\
\frac{\partial{V_{0}}}{\partial{\rho}}=\frac{\partial{V_{\rho}}}{\partial{x_{0}}},\end{cases}$
$\displaystyle\rho\left(\frac{\partial{{}^{2}}{g}}{\partial{x_{0}}^{2}}+\frac{\partial{{}^{2}}{g}}{\partial{\rho}^{2}}\right)+2\frac{\partial{g}}{\partial{\rho}}=0,\quad\quad\rho\left(\frac{\partial{{}^{2}}{\hat{g}}}{\partial{x_{0}}^{2}}+\frac{\partial{{}^{2}}{\hat{g}}}{\partial{\rho}^{2}}\right)-2\frac{\partial{\hat{g}}}{\partial{\rho}}=0.$

The characteristic equation $(\ref{characteristic lambda-4})$ in case
$\alpha=0$ is written as incomplete equation:

$\displaystyle\lambda^{3}+II_{\mathbf{J}(\vec{V})}\lambda-
III_{\mathbf{J}(\vec{V})}+IV_{\mathbf{J}(\vec{V})}\lambda=0,$ (6.2)

where

$\displaystyle II_{\mathbf{J}(\vec{V})}$
$\displaystyle=-\left[\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}\right]-\frac{E_{\rho}}{\rho}\frac{\partial{V_{\rho}}}{\partial{\rho}}-\left(\frac{V_{\rho}}{\rho}\right)^{2},$
$\displaystyle III_{\mathbf{J}(\vec{V})}$
$\displaystyle=-\frac{V_{\rho}}{\rho}\left[\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}\right]-\left(\frac{V_{\rho}}{\rho}\right)^{2}\frac{\partial{V_{\rho}}}{\partial{\rho}},$
$\displaystyle IV_{\mathbf{J}(\vec{V})}$
$\displaystyle=-\left(\frac{V_{\rho}}{\rho}\right)^{2}\left[\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}+2\frac{V_{\rho}}{\rho}\frac{\partial{V_{\rho}}}{\partial{\rho}}\right].$

######  Corollary 6.1.

Roots of the characteristic equation (6.2) are given by the formulas

$\displaystyle\lambda_{0,1}=\frac{V_{\rho}}{\rho},\;\lambda_{2,3}=-\frac{V_{\rho}}{\rho}\pm\sqrt{\left(\frac{V_{\rho}}{\rho}+\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}}.$
(6.3)

The set of degenerate points of the Jacobian matrix (6.1) is provided by two
independent equations:

$\displaystyle{V_{\rho}}=0,\quad\left(\frac{\partial{V_{\rho}}}{\partial{x_{0}}}\right)^{2}+\left(\frac{\partial{V_{\rho}}}{\partial{\rho}}\right)^{2}+2\frac{V_{\rho}}{\rho}\frac{\partial{V_{\rho}}}{\partial{\rho}}=0.$
(6.4)

Exact formulas (6.3), (6.4) demonstrate explicitly the geometric specifics of
the Jacobian matrix (6.1).

######  Definition 6.2.

Let $\Lambda\subset\mathbb{R}^{4}$ be a simply connected open domain, where
$x_{1}\neq 0,x_{2}\neq 0,x_{3}\neq 0$. Assume that an exact solution
$(u_{0},u_{1},u_{2},u_{3})$ of the system $(R_{4})$ satisfies axially
symmetric conditions
${u_{1}}{x_{2}}={u_{2}}{x_{1}},\quad{u_{1}}{x_{3}}={u_{3}}{x_{1}},\quad{u_{2}}{x_{3}}={u_{3}}{x_{2}}$
in $\Lambda$. Then mapping
$u=u_{0}+iu_{1}+ju_{2}+ku_{3}:\Lambda\rightarrow\mathbb{R}^{4}$ is called
four-dimensional harmonic meridional mapping of the first kind, while mapping
$\overline{u}=u_{0}-iu_{1}-ju_{2}-ku_{3}:\Lambda\rightarrow\mathbb{R}^{4}$ is
called four-dimensional harmonic meridional mapping of the second kind.

The set of degenerate points of every four-dimensional harmonic meridional
mapping of the second kind coincides with the set of degenerate points of the
Jacobian matrix (6.1).

Roots $\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{3}$ of the characteristic
equation (6.2) may be interpreted as the Poincaré stability coefficients in
the context of the theory of _Gradient dynamical systems with harmonic
potential_ , where $\vec{V}=\frac{d{\vec{x}}}{dt}=\mathrm{grad}\ h$, $\
\Delta\ h=0$. Positive, zero and negative values of the Poincaré stability
coefficients are of particular interest to _Stability theory_ (see, e.g., [12,
51, 38, 39]).

######  Example.

Consider a generalized axially symmetric potential in case $\alpha=0$ using
Bessel function of the first kind of order $-\frac{1}{2}$ and real argument
$\breve{\beta}\rho$:

$g(x_{0},\rho)=e^{\breve{\beta}x_{0}}\rho^{-\frac{1}{2}}J_{-\frac{1}{2}}(\breve{\beta}\rho).$

We deal with analytic models of harmonic potential meridional fields
$\vec{V}=(V_{0},\frac{x_{1}}{\rho}V_{\rho},\frac{x_{2}}{\rho}V_{\rho},\frac{x_{3}}{\rho}V_{\rho})$,
where

$\displaystyle
V_{0}=\breve{\beta}e^{\breve{\beta}x_{0}}\rho^{-\frac{1}{2}}J_{-\frac{1}{2}}(\breve{\beta}\rho),\quad
V_{\rho}=e^{\breve{\beta}x_{0}}\rho^{-\frac{1}{2}}\left(J^{\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)-\frac{1}{2\rho}J_{-\frac{1}{2}}(\breve{\beta}\rho)\right),$

such that

$\displaystyle\frac{\partial{V_{\rho}}}{\partial{x_{0}}}$
$\displaystyle=\breve{\beta}e^{\breve{\beta}x_{0}}\rho^{-\frac{1}{2}}\left(J^{\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)-\frac{1}{2\rho}J_{-\frac{1}{2}}(\breve{\beta}\rho)\right),$
$\displaystyle\frac{\partial{V_{\rho}}}{\partial{\rho}}$
$\displaystyle=e^{\breve{\beta}x_{0}}\rho^{-\frac{1}{2}}\left(J^{\prime\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)-\frac{1}{\rho}J^{\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)+\frac{3}{4\rho^{2}}J_{-\frac{1}{2}}(\breve{\beta}\rho)\right).$

Roots of the characteristic equation (6.2) are given by these formulas:

$\displaystyle\lambda_{0,1}=\frac{e^{\breve{\beta}x_{0}}}{\rho\sqrt{\rho}}\left(J^{\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)-\frac{1}{2\rho}J_{-\frac{1}{2}}(\breve{\beta}\rho)\right),$
$\displaystyle\lambda_{2,3}=-\frac{e^{\breve{\beta}x_{0}}}{\rho\sqrt{\rho}}\left(J^{\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)-\frac{1}{2\rho}J_{-\frac{1}{2}}(\breve{\beta}\rho)\right)\pm$
$\displaystyle\frac{e^{\breve{\beta}x_{0}}}{\sqrt{\rho}}\sqrt{\breve{\beta}^{2}\left(J^{\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)-\frac{1}{2\rho}J_{-\frac{1}{2}}(\breve{\beta}\rho)\right)^{2}+\left(J^{\prime\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)+\frac{1}{4\rho^{2}}J_{-\frac{1}{2}}(\breve{\beta}\rho)\right)^{2}}.$

The set of degenerate points of the Jacobian matrix (6.1) is provided by two
independent equations:

$\displaystyle
J^{\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)-\frac{1}{2\rho}J_{-\frac{1}{2}}(\breve{\beta}\rho)=0,$
$\displaystyle\left(\breve{\beta}^{2}-\frac{1}{\rho^{2}}\right)\left(J^{\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)-\frac{1}{2\rho}J_{-\frac{1}{2}}(\breve{\beta}\rho)\right)^{2}+\left(J^{\prime\prime}_{-\frac{1}{2}}(\breve{\beta}\rho)+\frac{1}{4\rho^{2}}J_{-\frac{1}{2}}(\breve{\beta}\rho)\right)^{2}=0.$

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