# On the notion of a quaternionic
holomorphic function

Michael Parfenov  Bashkortostan Branch of Russian Academy of Engineering, Ufa,
Russia

###### Abstract

A physically more adequate definition of a quaternionic holomorphic
(H-holomorphic) function of one quaternionic variable compared to known ones
and a quaternionic generalization of Cauchy-Riemann’s equations are presented.
At that a class of introduced H-holomorphic functions consists of those
quaternionic functions whose left and right derivatives become equal after the
transition to 3D space. The presented theory demonstrates a complete
similarity of the algebraic properties and differentiation rules between the
classes of H-holomorphic and ordinary complex holomorphic functions, including
the fact that quaternionic multiplication of the H-holomorphic functions
behaves as commutative and the fact that each H-holomorphic function can be
created from its complex holomorphic analogue by replacing a complex variable
by a quaternion one. A fairly large number of detailed examples are given to
illustrate the presented theory efficiency.

2020 Mathematics Subject Classification: 30G35.
Keywords: quaternionic differentiability, quaternionic holomorphic functions,
quaternionic analysis, quaternionic generalization of Cauchy-Riemann’s
equations.
Date: January 4, 2024
e-mail address<EMAIL_ADDRESS>
##  1 Introduction

The situation with regard to the derivative notion in quaternionic analysis
seems somewhat confused. It has even led R. Penrose to say [1, p. 201],
“…quaternions are not really so mathematically ‘nice’ as they seem at first
sigh. They are relatively poor ‘magicians’; and, certainly, they are no match
for complex numbers in this regard. The reason appears to be that there is no
satisfactory quaternionic analogue of the notion of a holomorphic function.”

During the last time there was a lot of interesting research (see, for
example, [2] \- [8]), including also new definitions of quaternionic
holomorphy, however they do apparently not provide a full correspondence of
properties of quaternionic holomorphic functions to complex analogues.

For a example, within the framework of existing conceptions, which consider
most often the left or right derivative in isolation, regarding another as
equivalent, we cannot construct a quaternionic holomorphic function from a
corresponding complex holomorphic function of the same type by a direct
replacement of a complex variable by a quaternion variable in an expression
for a complex holomorphic function although an analogous procedure is possible
(see, e.g., [9, p. 353]) when constructing complex holomorphic functions from
real-valued ones by a direct replacement of a real variable by a complex one.

In this regard, we’d like briefly to consider an essence of a simple approach
based on physically more adequate notion of a quaternionic holomorphic
function, which allows to call into question the above statement of R. Penrose
and to establish a new class of quaternionic holomorphic functions with
algebraic and differential properties fully similar to complex holomorphic
analogues. Although this article has an overview character we would like to
give a fairly large number of detailed examples illustrating the theory
efficiency.

##  2 Quaternionic holomorphic functions

The simple theory of quaternionic holomorphic functions can be built on
principles fully similar (essentially adequate) to ones of complex holomorphic
functions. At that the general definition of a derivative is based on the
following main idea, viz.: each point of any real line is at the same time a
point of some plane and 3D space as a whole, and therefore any
characterization of differentiability (and its relations) at a point must be
the same regardless of whether we think of that point as a point on the real
axis or a point in the complex plane or a point in three-dimensional space. It
follows that a quaternionic derivative of a quaternionic function
$\psi\left(p\right)$ must be defined similar to complex derivative as a limit
of a difference quotient of $\triangle\psi\left(p\right)$ by $\triangle p$
when $\triangle p$ converges to zero along any direction in the quaternionic
space, where
$\psi\left(p\right)=\psi_{1}\left(x,y,z,u\right)+\psi_{2}\left(x,y,z,u\right)i+\psi_{3}\left(x,y,z,u\right)j+\psi_{4}\left(x,y,z,u\right)k$
is a quaternionic function of real components $x,y,z,u$ of an independent
quaternionic variable $p=x+yi+zj+uk$. The functions
$\psi_{1}\left(x,y,z,u\right),\psi_{2}\left(x,y,z,u\right),\psi_{3}\left(x,y,z,u\right),\psi_{4}\left(x,y,z,u\right)$
are real-valued and $i,j,k$ are the base quaternions of the quaternion space
$\mathbb{H}$.

Using the Cayley–Dickson construction (doubling form) [10] we have

$\begin{gathered}p=a+b\cdot
j\in\mathbb{H},\;\;\;\;\;\;\psi\left(p\right)=\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)+\Phi_{2\cdot}\left(a,b,\overline{a},\overline{b}\right)\cdot
j\in\mathbb{H},\\\ \,\,\,\overline{p}=\overline{a}-b\cdot
j\subset\mathbb{H}\text{,}\;\;\;\;\;\;\overline{\psi\left(p\right)}=\overline{\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)}\;-\Phi_{2\cdot}\left(a,b,\overline{a},\overline{b}\right)\cdot
j\subset\mathbb{H}\text{,}\end{gathered}$ (1)

where

$\begin{gathered}a=x+yi,\;\;\;\;\;b=z+ui,\\\
\overline{a}=x-yi,\;\;\;\;\;\overline{b}=z-ui,\end{gathered}$ (2)
$\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)=\psi_{1}+\psi_{2}i,\;\;\;\;\;\Phi_{2}\left(a,b,\overline{a},\overline{b}\right)=\psi_{3}+\psi_{4}i,$
$\overline{\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)}=\psi_{1}-\psi_{2}i,\;\;\;\;\overline{\Phi_{2}\left(a,b,\overline{a},\overline{b}\right)}=\psi_{3}-\psi_{4}i$

are compex quantities, the “$\cdot$” and overbar signs denote, respectively,
quaternionic multiplication and complex (or quaternionic if needed)
conjugation. For simplicity, we use also the short designations $\Phi_{1}$ and
$\Phi_{2}$ respectively instead of
$\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)$ and
$\Phi_{2}\left(a,b,\overline{a},\overline{b}\right)$.

Since the quaternion algebra is a noncommutative algebra with division [10],
there can exist the left and the right definition of a quaternionic
derivative:

$\psi_{left}^{\prime}=\lim_{\triangle p\rightarrow 0}\left[\left(\triangle
p\right)^{-1}\cdot\left\\{\psi\left(p\,+\triangle
p\right)-\psi\left(p\right)\right\\}\right]\\!,$
$\,\,\,\psi_{right}^{\prime}=\lim_{\triangle p\rightarrow
0}\left[\left\\{\psi\left(p\,+\triangle
p\right)-\psi\left(p\right)\right\\}\cdot\left(\triangle
p\right)^{-1}\right]\\!.$

Usually, one of them is used (regarding another as equivalent [2, 3], etc).
However such an approach cannot be recognized as full, since the algebra
underlying the quaternionic analysis needs at the same time to use both
quaternionic multiplications: on the left and on the right (see details in
[10, p. 32] and [11, p. 7]) to represent all arbitrary (and
noncommutative) rotations of any vector in 3D space just like as the complex
algebra underlying the complex analysis uses the complex multiplication,
representing all arbitrary (commutative) rotations of any vector in the
complex plane. Thus we have to refuse to consider only the left or only the
right approach (regarding another as equivalent) when defining a quaternionic
derivative. The left and right approaches should be considered only together
[11].

In complex analysis the notion of a derivative of a holomorphic function (as a
complex potential) is always associated [9, 12] with an unambiguous strength
value of some physical planar vector field (electrical field, fluid flow and
etc.) and accordingly the derivative is unambiguous. Similarly, the
quaternionic derivative must correspond some physical field of dimension 3 and
hence must be also unambiguous. Thus we are forced to require an equality of
the left and right derivatives,
i.e.$\;\psi_{left}^{\prime}=\psi_{right}^{\prime}$ in a domain of a function
definition to associate the quaternionic derivative with physical reality of
3D space.

In fact, we want to define the notion of a quaternionic differentiability,
which will be “physically more adequate ” (or essentially adequate) than the
left or the right approach in isolation. At first sight the requirement
$\psi_{left}^{\prime}=\psi_{right}^{\prime}$ taken alone seems absurd, but we
will try to establish those quaternionic functions and those conditions for
which it is fair. As we see below, such the functions are desired quaternionic
holomorphic functions and such the conditions are quaternionic Cauchy-
Riemann’s equations that are performed for these functions after the
transition to 3D space. At that the left and the right derivatives of such
functions become equal after the transition to 3D space.

According to these principles, it follows that the limit of the difference
quotient is required to be independent not only of directions to approach a
limiting point when $\triangle p\rightarrow 0$ (as in complex analysis [12,
9]), but also of the manner of quaternionic division: on the left or on the
right. In this sence we can say that “the derivative is independent of the way
of its computation ”. Based on that, it is possible to introduce the following

Definition 1. A single-valued quaternionic function
$\psi\left(p\right):G_{4}\rightarrow\mathbb{H}$ is quaternionic differentiable
at a point $p\in G_{4}\subseteq\mathbb{H}$ if there exists a limiting value of
the difference quotient ${\triangle\psi\left(p\right)}/{\triangle p}$ as
$\triangle p\rightarrow 0,$ and this value is independent of the way of its
computation.

As we shall see further, such an independence is provided after the transition
to 3D space.

Definition 2. A quaternionic function is said to be a quaternionic holomorphic
function (briefly, H-holomorphic function) at a point $p\in\mathbb{H}$ if it
has a quaternionic derivative independent of a way of its computation in some
open connected neighborhood $G_{4}\subset\mathbb{H}$ of a point $p$.

In the Cayley–Dickson doubling form Definitions 1 and 2 (by equating the left
and the right derivatives) lead to the following formulation of the necessary
and sufficient conditions [11, p.p. 14-19, p. 22] for the function
$\psi\left(p\right)$ to be H-holomorphic:

Definition 3. It is assumed that the constituents
$\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)$ and
$\Phi_{2}\left(a,b,\overline{a},\overline{b}\right)$ of a quaternionic
function
$\psi\left(p\right)=\psi\left(a,b,\overline{a},\overline{b}\right)=\Phi_{1}+\Phi_{2}j$
possess continuous first-order partial derivatives with respect to
$a,b,\overline{a}$ and $\overline{b}$ in some open connected neighborhood
$G_{4}\subset\mathbb{H}$ of a point $p\in\mathbb{H}.$ Then a function
$\psi\left(p\right)$ is said to be H-holomorphic and denoted by
$\psi_{H}\left(p\right)$ at a point $p,$ if and only if the functions
$\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)$ and
$\Phi_{2}\left(a,b,\overline{a},\overline{b}\right)$ satisfy in $G_{4}$ the
following quaternionic generalization of complex Cauchy-Riemann’s equations
[11]:

$\left\\{\begin{aligned}
1)\,\,\,\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\,\partial_{\overline{b}}\overline{\Phi_{2}}\\!\mid,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2)\,\,\,\,(\,\partial_{a}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\overline{\Phi_{1}}\\!\mid,\\\
3)\,\,\,\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\,\partial_{b}\Phi_{2}\\!\mid,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4)\,\,\,\,(\,\partial_{\overline{a}}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\Phi_{1}\\!\\!\mid.\end{aligned}\right.$
(3)

Here $\partial_{i},\,i=a,\overline{a},b,\overline{b}$ denotes the partial
derivative with respect to $i$. The brackets $\left(\cdots{}\\!\\!\mid\right.$
with the closing vertical bar indicate that the transition $a=\overline{a}=x$
(to 3D space) has been already performed in expressions enclosed in brackets.
Equations (3-1) and (3-2) have the components relating to the left
quaternionic derivative and equations (3-3) and (3-4) to the right one [11,
p.p. 16,17]. The requirement $a=\overline{a}=x$ provides the possibility of
joint implementation of equations (3-1,2) for the left quaternionic derivative
and (3-3,4) for the right one. Equations (3-1) and (3-3) as well as (3-2) and
(3-4) become, respectively, identical after the transition to 3D space [11,
p. 18], i.e. the left derivative becomes equal to the right one.

We see that H-holomorphy conditions (3) are defined so that during the check
of the quaternionic holomorphy of any quaternionic function we have to do the
transition $a=\overline{a}=x$ in already calculated expressions for the
partial derivatives of the functions
$\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)$ and
$\Phi_{2}\left(a,b,\overline{a},\overline{b}\right)$ and their complex
conjugations. However, this doesn’t mean that we deal with triplets in
general, since the transition $a=\overline{a}=x$ (or $y=0$) cannot be
initially done for quaternionic variables and functions. Any quaternionic
function remains the same 4-dimensional quaternionic function regardless of
whether we check its holomorphy or not. Simply put, the H-holomorphic
functions are 4-dimensional quaternionic functions for which the partial
derivatives of components of the Cayley–Dickson doubling form satisfy
equations (3) after the transition to 3D space. In other words, they are those
quaternionic functions whose the left and the right derivatives become equal
after the transition to 3D space. That’s not surprising, since unambiguous
stationary physical fields, represented by derivatives, exist precisely in 3D
space.

###  2.1 Example of $p$ to the power of 2

The function $\psi\left(p\right)=p^{2}=\left(a+b\cdot
j\right)\cdot\left(a+b\cdot
j\right)=\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)+\Phi_{2}\left(a,b,\overline{a},\overline{b}\right)\cdot
j.\,\,$By direct quaternionic multiplication we obtain the following
expressions for the components of the Cayley–Dickson doubling form:

$\Phi_{1}\\!\\!\left(a,b,\overline{a},\overline{b}\right)=a^{2}-b\overline{b}\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\Phi_{2}\\!\\!\left(a,b,\overline{a},\overline{b}\right)=\left(a+\overline{a}\right)b.$

Correspondingly, the complex conjugate functions are

$\,\,\overline{\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)}=\overline{a}^{2}-\overline{b}b\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\overline{\Phi_{2}\left(a,b,\overline{a},\overline{b}\right)}=\left(\overline{a}+a\right)\overline{b}.$

When extracting the components
$\Phi_{1}\\!\\!\left(a,b,\overline{a},\overline{b}\right)$ and
$\Phi_{2}\\!\\!\left(a,b,\overline{a},\overline{b}\right)$ we always take into
consideration the fact that $j\alpha=\overline{\alpha}j$ [10, p. 42] for any
$\alpha\in\mathbb{C}$, where $\mathbb{C}$ is the complex plane and $j$ is the
imaginary quaternion unit, so its square is -1.

Now we calculate the derivatives:
$\partial_{a}\Phi_{1}=2a,\,\,\,\partial_{\overline{b}}\overline{\Phi_{2}}=\overline{a}+a,\,\,\,\partial_{a}\Phi_{2}=\leavevmode\nobreak\
b,\,\,\,\partial_{\overline{b}}\overline{\Phi_{1}}=-b$ for equations (3-1,2)
and
$\,\,\,\partial_{b}\Phi_{2}=a+\overline{a},\,\,\,\partial_{\overline{a}}\Phi_{2}=b,\,\,\,\partial_{\overline{b}}\Phi_{1}=-b$
for equations (3-3,4). By substituting $a=\overline{a}$, i.e. $a=x$ and
$\overline{a}=x$ into these expressions for partial derivatives we have

$\left\\{\begin{aligned}
1)\,\,\,\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\,\partial_{\overline{b}}\overline{\Phi_{2}}\\!\mid=2x,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2)\,\,\,\,(\,\partial_{a}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\overline{\Phi_{1}}\\!\mid=b,\\\
3)\,\,\,\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\,\partial_{b}\Phi_{2}\\!\mid=2x,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4)\,\,\,\,(\,\partial_{\overline{a}}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\Phi_{1}\\!\\!\mid=b\end{aligned}\right.$

in coordinates $x,z,u$ of 3D space. We see that equations (3) are fulfilled,
i.e. the 4-dimensional quaternionic function $\psi\left(p\right)=p^{2}$ is
H-holomorphic in $\mathbb{H}$; equations (3-1) and (3-3) as well as equations
(3-2) and (3-4) are correspondingly identical, and hence the left quaternionic
derivative equals the right one after the transition to 3D space.

###  2.2 Example of natural exponential function

The function $\psi\left(p\right)=e^{p}=\Phi_{1}+\Phi_{2}\cdot j,$ where $e$ is
the base of the natural logarithm. We represent the quaternion variable
$p=x+yi+zj+uk$ as a sum of real and imaginary parts: $p=x+vr,$ where
$v=\sqrt{y^{2}+z^{2}+u^{2}}$ is a real value and

$r=\frac{yi+zj+uk}{\sqrt{y^{2}+z^{2}+u^{2}}}$ (4)

is a purely imaginary unit quaternion, so its square is $-1.$ We will consider
further $v>0.$ Since $r^{2}=-1$ as well as $x$ and $v$ are real values, the
quaternionic formula $p=x+vr$ is algebraically equivalent to the complex
formula $\xi=x+yi$ (here and further, if needed, we use the designation $\xi$
for a complex variable instead of the usual in complex analysis designation
$z$ in order to avoid confusing with the quaternionic component $z$). Then,
using the quaternionic analogue of Euler’s formula: $e^{vr}=\cos v+r\sin v,$
we have

$\psi\left(p\right)=\Phi_{1}+\Phi_{2}\cdot
j=e^{p}=e^{\left(x+vr\right)}=e^{x}e^{vr}=e^{x}\left(\cos v+r\sin v\right)$
$=e^{x}\left(\cos v+\frac{yi\sin v}{v}\right)+e^{x}\frac{\left(z+ui\right)\sin
v}{v}\cdot j,$

whence

$\Phi_{1}\\!=e^{x}\left(\cos v+\frac{yi\sin
v}{v}\right)\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\Phi_{2}=e^{x}\frac{\left(z+ui\right)\sin
v}{v}.$

Using the expressions
$x=\frac{a+\overline{a}}{2},\,y=\frac{a-\overline{a}}{2i},\,z=\frac{b+\overline{b}}{2},\,u=\frac{b-\overline{b}}{2i},$
following from (2), we obtain the expressions for $\Phi_{1}$ and $\Phi_{2}$ as
functions of $a,\,\overline{a},\,b,\,\overline{b}:$

$\Phi_{1}\\!=2\beta\cos v+\frac{\beta\left(a-\overline{a}\right)\sin
v}{v}\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,\Phi_{2}=\frac{2\beta\,b\sin
v}{v},$

where

$\begin{gathered}\beta=\frac{e^{\frac{a+\overline{a}}{2}}}{2}=\overline{\beta},\,\,\,\,\,\,v=\sqrt{y^{2}+|b|^{2}}=\frac{\sqrt{\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}}{2}=\overline{v},\\\
|p|=\sqrt{x^{2}+y^{2}+z^{2}+u^{2}}=\sqrt{a\overline{a}+b\overline{b}}\end{gathered}$
(5)

are real values. We see that in fact we have obtained the expression for
quaternionic function $\psi\left(p\right)=e^{p}$ from the expression for
complex function $\psi\left(\xi\right)=e^{\xi}$ by the direct replacement of a
complex variable $\xi=x+yi$ by a quaternion variable $p=x+vr.$ Since the
complex imaginary unit $i$ must be replaced by its quaternionic analogue $r$,
we need to provide for such a replacement everywhere when constructing
quaternionic holomorphic functions.

Over 40 various H-holomorphic functions, obtained by the replacement of a
complex variable by a quaternionic one in expressions for initial complex
holomorphic functions, are represented in [13].

The partial derivatives of $\Phi_{1}$ and $\Phi_{2}$ for the function $e^{p}$
are the following:

$\displaystyle\partial_{a}\Phi_{1}=\beta\left[\cos
v+\frac{\left(a-\overline{a}+1\right)\sin
v}{v}-\frac{\left(a-\overline{a}\right)^{2}\left(v\cos v-\sin
v\right)}{4v^{3}}\right],$
$\displaystyle\partial_{b}\Phi_{2}=\partial_{\overline{b}}\overline{\Phi_{2}}=\beta\left[\frac{2\sin
v}{v}+\frac{b\overline{b}\left(v\cos v-\sin v\right)}{v^{3}}\right],$
$\displaystyle\partial_{a}\Phi_{2}=-\partial_{\overline{b}}\Phi_{1}=\beta
b\left[\frac{\sin v}{v}-\frac{\left(a-\overline{a}\right)\left(v\cos v-\sin
v\right)}{2v^{3}}\right],$
$\displaystyle\partial_{\overline{a}}\Phi_{2}=-\partial_{\overline{b}}\overline{\Phi_{1}}=\beta
b\left[\frac{\sin v}{v}+\frac{\left(a-\overline{a}\right)\left(v\cos v-\sin
v\right)}{2v^{3}}\right].$

Performing the transition $a=\overline{a}=x$ in them and taking into
consideration that $v=|b|>0$ and $\beta=\frac{e^{x}}{2}$ after this transition
as well as $b\overline{b}=|b|^{2}$ we get

$\left\\{\begin{aligned}
1)\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\partial_{\overline{b}}\overline{\Phi_{2}}\\!\mid=\frac{e^{x}\left(\cos|b|+|b|^{-1}\sin|b|\right)}{2}\\!;\,\,2)\,(\,\partial_{a}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\overline{\Phi_{1}}\\!\mid=\frac{e^{x}b|b|^{-1}\sin|b|}{2}\\!;\\\
3)\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\partial_{b}\Phi_{2}\\!\mid=\frac{e^{x}\left(\cos|b|+|b|^{-1}\sin|b|\right)}{2}\\!;\,\,4)\,(\,\partial_{\overline{a}}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\Phi_{1}\\!\\!\mid=\frac{e^{x}b|b|^{-1}\sin|b|}{2}\\!.\end{aligned}\right.$

We see that equations (3) are fulfilled and the function
$\psi\left(p\right)=e^{p}$ is H-holomorphic everywhere in $\mathbb{H}$,
including the point $p=0$. Indeed, at this point we have equations (3) as
follows:

$\left\\{\begin{aligned}
1)\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\partial_{\overline{b}}\overline{\Phi_{2}}\\!\mid=1;\,\,2)\,(\,\partial_{a}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\overline{\Phi_{1}}\\!\\!\mid=0;\\\
3)\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\partial_{b}\Phi_{2}\\!\mid=1;\,\,4)\,(\,\partial_{\overline{a}}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\Phi_{1}\\!\\!\mid=0,\end{aligned}\right.$

which are also fulfilled.

Equations 1) and 3) as well as 2) and 4) are respectively identical and hence
the left quaternionic derivative is equal to the right one after the
transition $a=\overline{a}=x.$

##  3 The full quaternionic derivative

It was established in [11] that the quaternionic generalization of the complex
derivative has the following expression for the full (uniting the parts of
left and right derivatives [15, p. 18]) quaternionic derivative of the
$k^{\prime}\text{th}$ order:

$\psi_{H}^{\left(k\right)}\\!\\!\left(p\right)=\Phi_{1}^{(k)}+\Phi_{2}^{(k)}\cdot
j,$ (6)

where the constituents $\Phi_{1}^{(k)}$ and $\Phi_{2}^{(k)}$ are expressed by

$\Phi_{1}^{(k)}=\partial_{a}\Phi_{1}^{(k-1)}\\!\\!+\partial_{\overline{a}}\Phi_{1}^{(k-1)}\,\,\,\text{and}\,\,\,\,\,\,\,\Phi_{2}^{(k)}=\partial_{a}\Phi_{2}^{(k-1)}\\!\\!+\partial_{\overline{a}}\Phi_{2}^{(k-1)},$

and $\Phi_{1}^{(k-1)}\text{and}\,\,\Phi_{2}^{(k-1)}$ are the constituents of
the $\left(k-1\right)^{{}^{\prime}}\\!\text{th}$ full derivative of
$\psi_{H}\left(p\right)$ represented in the Cayley–Dickson doubling form as
$\psi_{H}^{\left(k-1\right)}\\!\\!\left(p\right)=\Phi_{1}^{(k-1)}+\Phi_{2}^{(k-1)}\cdot
j,k\geq 1;$
$\Phi_{1}^{(0)}=\Phi_{1}\\!\\!\left(a,b,\overline{a},\overline{b}\right)\text{and}\,\,\Phi_{2}^{(0)}=\Phi_{2}\\!\\!\left(a,b,\overline{a},\overline{b}\right).$

If a quaternion function $\psi\left(p\right)$ is once H-differentiable in
$G_{4}\subset\mathbb{H},$ then it possesses the full quaternionic derivatives
of all orders in $G_{4}$ (except, possibly, at certain singularities), each
one H-differentiable [11]. Analogously to complex analysis there are various
quaternionic equivalents of formula (6). See details in [11, 15].

The full first quaternionic derivative of the function
$\psi_{H}\left(p\right)=p^{2}$ (see Example 2.1 ) is the following:

$\displaystyle\psi_{H}^{(1)}\\!\\!\left(p\right)$
$\displaystyle=\left(p^{2}\right)^{{\left(1\right)}}=\Phi_{1}^{(1)}+\Phi_{2}^{(1)}\cdot
j=\left(\partial_{a}\Phi_{1}\\!+\partial_{\overline{a}}\Phi_{1}\right)+\left(\partial_{a}\Phi_{2}\\!+\partial_{\overline{a}}\Phi_{2}\right)\cdot
j$ $\displaystyle=\left(2a+0\right)+\left(b+b\right)\cdot j=2a+2b\cdot j=2p.$

Given that the function $e^{p}$ has
$\partial_{\overline{a}}\Phi_{1}=\beta\left[\frac{v\cos v-\sin
v}{v}+\frac{\left(a-\overline{a}\right)^{2}\left(v\cos v-\sin
v\right)}{4v^{3}}\right],$ we obtain after some computation the expression for
the first full quaternionic derivative of the function
$\psi_{H}\left(p\right)=e^{p}$ (see Example 2.2) as follows:

$\begin{split}\psi_{H}^{(1)}\\!\\!\left(p\right)&=\left(e^{p}\right)^{{\left(1\right)}}=\Phi_{1}^{(1)}+\Phi_{2}^{(1)}\cdot
j=\left(\partial_{a}\Phi_{1}\\!+\partial_{\overline{a}}\Phi_{1}\right)+\left(\partial_{a}\Phi_{2}\\!+\partial_{\overline{a}}\Phi_{2}\right)\cdot
j\\\ &=\left\\{\beta\left[\cos v+\frac{\left(a-\overline{a}+1\right)\sin
v}{v}-\frac{\left(a-\overline{a}\right)^{2}\left(v\cos v-\sin
v\right)}{4v^{3}}\right]\right.\\\
&\left.\,\,\,\,\,\,\,\,\,\,\,\,+\beta\left[\frac{v\cos v-\sin
v}{v}+\frac{\left(a-\overline{a}\right)^{2}\left(v\cos v-\sin
v\right)}{4v^{3}}\right]\right\\}\\\ &+\left\\{\beta b\left[\frac{\sin
v}{v}-\frac{\left(a-\overline{a}\right)\left(v\cos v-\sin
v\right)}{2v^{3}}\right]\right.\\\ &\left.\,\,\,\,\,\,\,\,\,\,\,\,+\beta
b\left[\frac{\sin v}{v}+\frac{\left(a-\overline{a}\right)\left(v\cos v-\sin
v\right)}{2v^{3}}\right]\right\\}\cdot j\\\ &=2\beta\cos
v+\frac{\beta\left(a-\overline{a}\right)\sin v}{v}+\frac{2\,\beta\,b\sin
v}{v}\cdot j=e^{p}.\end{split}$

We see that the expressions for the full first quaternionic derivatives of the
H- holomorphic functions $\psi_{H}\\!\left(p\right)=p^{2}$ and
$\psi_{H}\\!\left(p\right)=e^{p}$ have the same forms as in real and complex
analysis. The same there is for the derivatives of higher orders. We also see
that the full quaternionic derivatives of all orders of the natural
exponential function are equal to the function itself just as in complex (and
real) analysis.

##  4 The new class of H-holomorphic functions

We introduce a new class of H-holomorphic functions, which consists of the
quaternionic functions meeting equations (3). In accordance with [14] we point
out here some properties of this class :

1) Each H-holomorphic function $\psi_{H}\left(p\right)$ can be constructed
from its complex holomorphic analogue $\psi_{C}\left(\xi\right)$ (without
change of a functional dependence form) by replacing a complex variable
$\xi\in G_{2}\subseteq\mathbb{C}$ as a single whole in an expression for
$\psi_{C}\left(\xi\right)$ by a quaternionic variable $p\in
G_{4}\subseteq\mathbb{H},$ where $G_{4}$ is defined (except, possibly, at
certain singularities) by the relation $G_{4}\supset G_{2}$ in the sense that
$G_{2}$ exactly follows from $G_{4}$ upon the transition from $p$ to $\xi$
[14, p. 15]. At that we must first convert the initial expression for a
complex holomorphic function into the expression in which there is dependence
on complex variable (and/or its conjugation) only as a single whole.

2) The constituents of the H-holomorphic functions (and their derivatives of
all orders) in the Cayley-Dickson doubling form (1) have the following general
representation forms [15]:

$\Phi_{1}\\!\\!\left(a,b,\overline{a},\overline{b}\right)=A\left[a,\overline{a},\left(b\overline{b}\right)\\!\right]\\!,\,\Phi_{2}\\!\\!\left(a,b,\overline{a},\overline{b}\right)=B\\!\left[\left(a\overline{a}\right)\\!,\left(a\overline{a}\right)_{m}\\!,\left(b\overline{b}\right)\right]\\!\\!b=Bb,\,B=\overline{B},$

where $\left(a\overline{a}\right)_{m}$ denotes the form
$a^{m}\overline{a}^{0}+a^{{\left(m-1\right)}}\overline{a}^{1}+a^{{\left(m-2\right)}}\overline{a}^{2}+\cdots+a^{0}\overline{a}^{m},m=0,1,2,3,\dots$
with the number of summands $\left(m+1\right).$ Together with this form or
instead of it, there can be another symmetric forms that are invariant under
complex conjugation [15, 11]. Such the forms of $\Phi_{1}$ and $\Phi_{2}$ are
typical of H-holomorphic functions and their derivatives. They can serve as
gauge for correctness of results obtained when constructing H-holomorphic
functions. Such a symmetry in cases of derivatives is a consequence of uniting
unsymmetrical parts of the left and right derivatives [15]. All examples in
this article illustrate such the forms (see also [13]) for H-holomorphic
functions and their full quaternionic derivatives.

3) Algebraic properties of this class are all identical to ones of the class
of complex holomorphic functions. In particular, the quaternionic
multiplication of the H-holomorphic functions behaves as commutative and the
left quotient of two H-holomorphic functions is equal to the right one [14].

4) The formulae for differentiating sums, products, ratios, inverses, and
compositions of H-holomorphic functions are all identical to their complex and
real analogues [14]. The expressions for the full quaternionic derivatives of
the H-holomorphic functions are all identical to the expressions for
corresponding derivatives of complex holomorphic analogues. For example, if
the first derivative of the complex holomorphic function
$\psi_{C}\left(\xi\right)=\xi^{3}$ is $3\xi^{2},$ then the first full
quaternionic derivative of the H-holomorphic analogue
$\psi_{H}\left(p\right)=p^{3}$ is $3p^{2}$ [11, 14]. One can just verify these
properties, constructing H-holomorphic functions from their complex
holomorphic analogues. The H-holomorphic functions possess the full
H-holomorphic quaternionic derivatives of all orders. We can denote the
presented class of H-holomorphic functions by $\text{H}^{\infty}.$ Note that
the known Cauchy-Riemann equations [12] follow from their quaternionic
generalization (3) upon the transition to complex plane [11, p. 29].

5) The presented concept of quaternionic holomorphy allows us to explore
steady state vector fields in 3D space, each of them corresponds to some
H-holomorphic function considered as a quaternionic potential [16].

###  4.1 Example of reciprocal function

The reciprocal function $\psi\left(p\right)=p^{{-1}},\,\,\,p\neq 0.$ The
corresponding complex holomorphic analogue is
$\psi\left(\xi\right)=\xi^{{-1}}=\frac{\overline{\xi}}{|\xi|^{2}},\,\,\,\xi\neq
0.$ By replacing a complex variable $\xi$ by a quaternionic $p$ in it we
obtain the following expression for the H-holomorphic function:
$\psi\left(p\right)=p^{{-1}}=\frac{\overline{p}}{|p|^{2}}=\frac{\overline{a}}{\left(a\overline{a}+b\overline{b}\right)}-\frac{b}{\left(a\overline{a}+b\overline{b}\right)}\cdot
j=\Phi_{1}+\Phi_{2}\cdot j,\,\,\,\,p\neq 0,$ whence
$\Phi_{1}=\frac{\overline{a}}{\left(a\overline{a}+b\overline{b}\right)}$ and
$\Phi_{2}=-\frac{b}{\left(a\overline{a}+b\overline{b}\right)}.$
Correspondingly,
$\overline{\Phi_{1}}=\frac{a}{\left(a\overline{a}+b\overline{b}\right)}$ and
$\overline{\Phi_{2}}=-\frac{\overline{b}}{\left(a\overline{a}+b\overline{b}\right)}.$

To check the H-holomorphy of the function $\psi\left(p\right)=p^{{-1}}$ we
compute the partial derivatives of the functions $\Phi_{1}$ and
$\Phi_{2}\\!\\!:\partial_{a}\Phi_{1}=-\frac{\overline{a}^{2}}{\left(a\overline{a}+b\overline{b}\right)^{2}},\,\,\,\partial_{\overline{b}}\overline{\Phi_{2}}=\partial_{b}\Phi_{2}=-\frac{a\overline{a}}{\left(a\overline{a}+b\overline{b}\right)^{2}},\,$
and
$\partial_{a}\Phi_{2}=-\partial_{\overline{b}}\Phi_{1}=\frac{\overline{a}b}{\left(a\overline{a}+b\overline{b}\right)^{2}},\,\,\,\partial_{\overline{a}}\Phi_{2}=-\partial_{\overline{b}}\overline{\Phi_{1}}=\frac{ab}{\left(a\overline{a}+b\overline{b}\right)^{2}}.$
After performing the transition $a=\overline{a}=x$ we see that H-holomorphy
equations (3) are fulfilled:

$\left\\{\begin{aligned}
1)\,\,\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\,\partial_{\overline{b}}\overline{\Phi_{2}}\\!\mid=-\frac{x^{2}}{\left(x^{2}+b\overline{b}\right)^{2}},\,\,\,\,\,\,\,\,\,2)\,\,\,(\,\partial_{a}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\overline{\Phi_{1}}\\!\mid=\frac{xb}{\left(x^{2}+b\overline{b}\right)^{2}},\\\
3)\,\,\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\,\partial_{b}\Phi_{2}\\!\mid=-\frac{x^{2}}{\left(x^{2}+b\overline{b}\right)^{2}},\,\,\,\,\,\,\,\,\,4)\,\,\,(\,\partial_{\overline{a}}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\Phi_{1}\\!\\!\mid=\frac{xb}{\left(x^{2}+b\overline{b}\right)^{2}}.\end{aligned}\right.$

The function $\psi\left(p\right)=p^{{-1}}$ is H-holomorphic at the points
$p\in\\!\mathbb{H}\\!\setminus\\!\\!\left\\{0\right\\}$ and its left and right
quaternionic derivatives become equal after the transition $a=\overline{a}=x$
to 3D space.
The first derivative of the function $\psi_{H}\\!\left(p\right)=p^{{-1}}$.
Using the expressions for $\partial_{a}\Phi_{1},\,\partial_{a}\Phi_{2},\\\
\partial_{\overline{a}}\Phi_{2}$ as well as
$\partial_{\overline{a}}\Phi_{1}=\frac{b\overline{b}}{\left(a\overline{a}+b\overline{b}\right)^{2}}$
in (6), we get the following expression for the first full (further we omit,
for brevity, the word “ full”) quaternionic derivative of the function
$\psi_{H}\\!\left(p\right)=p^{{-1}}\\!:\\\ $

$\displaystyle\left(p^{{-1}}\right)^{{\left(1\right)}}=\Phi_{1}^{(1)}+\Phi_{2}^{(1)}\cdot
j=\left(\partial_{a}\Phi_{1}\\!+\partial_{\overline{a}}\Phi_{1}\right)+\left(\partial_{a}\Phi_{2}\\!+\partial_{\overline{a}}\Phi_{2}\right)\cdot
j$
$\displaystyle=\left[-\frac{\overline{a}^{2}}{\left(a\overline{a}+b\overline{b}\right)^{2}}+\frac{b\overline{b}}{\left(a\overline{a}+b\overline{b}\right)^{2}}\right]+\left[\frac{\overline{a}\,b}{\left(a\overline{a}+b\overline{b}\right)^{2}}+\frac{a\,b}{\left(a\overline{a}+b\overline{b}\right)^{2}}\right]\cdot
j$
$\displaystyle=-\frac{\left[\overline{a}^{2}-b\overline{b}-b\left(a+\overline{a}\right)\cdot
j\right]}{\left(a\overline{a}+b\overline{b}\right)^{2}}=-\frac{\overline{p}^{2}}{\left(a\overline{a}+b\overline{b}\right)^{2}}=-\frac{\overline{p}^{2}}{|p|^{4}}.$

Given $p^{{-2}}\\!=p^{{-1}}\cdot p^{{-1}}=\frac{\overline{p}^{2}}{|p|^{4}},$
we get the following expression for the first derivative of the function
$\psi_{H}\left(p\right)=p^{{-1}}\\!:$

$\left(p^{{-1}}\right)^{{\left(1\right)}}=\Phi_{1}^{(1)}+\Phi_{2}^{(1)}\cdot
j=-p^{{-2}},$ (7)

where
$\Phi_{1}^{(1)}=\frac{b\overline{b}-\overline{a}^{2}}{{\left(a\overline{a}+b\overline{b}\right)^{2}}},\,\Phi_{2}^{(1)}=\frac{b\left(a+\overline{a}\right)}{{\left(a\overline{a}+b\overline{b}\right)^{2}}}.$
Their complex conjugations are
$\overline{\Phi_{1}^{(1)}}=\frac{\overline{b}b-a^{2}}{{\left(a\overline{a}+b\overline{b}\right)^{2}}},\\\
\overline{\Phi_{2}^{(1)}}=\frac{\overline{b}\left(\overline{a}+a\right)}{{\left(a\overline{a}+b\overline{b}\right)^{2}}}.$
We see that the obtained expression (7) for the first quaternionic derivative
of the function $p^{{-1}}$ is similar to the expression for the complex
analogue: $\left(\xi^{{-1}}\right)^{{\left(1\right)}}=-\xi^{{-2}}.$

To verify the H-holomorphy of the first derivative
$\left(p^{{-1}}\right)^{{\left(1\right)}}$ we calculate the following partial
derivatives:
$\partial_{a}\Phi_{1}^{(1)}=\frac{2\overline{a}\left(\overline{a}^{2}-b\overline{b}\right)}{\left(a\overline{a}+b\overline{b}\right)^{3}},\,\partial_{b}\Phi_{2}^{(1)}=\partial_{\overline{b}}\overline{\Phi_{2}^{(1)}}=\frac{\left(a+\overline{a}\right)\left(a\overline{a}-b\overline{b}\right)}{\left(a\overline{a}+b\overline{b}\right)^{3}}$
and
$\partial_{a}\Phi_{2}^{(1)}=-\partial_{\overline{b}}\Phi_{1}^{(1)}=\frac{b\left(b\overline{b}-a\overline{a}-2\overline{a}^{2}\right)}{\left(a\overline{a}+b\overline{b}\right)^{3}},\,\partial_{\overline{a}}\Phi_{2}^{(1)}=-\partial_{\overline{b}}\overline{\Phi_{1}^{(1)}}=\frac{b\left(b\overline{b}-a\overline{a}-2a^{2}\right)}{\left(a\overline{a}+b\overline{b}\right)^{3}}.$
After performing the transition $a=\overline{a}=x$ we see that the
H-holomorphy equations (3) are fulfilled for the first derivative of the
function $\psi_{H}\\!\left(p\right)=p^{{-1}}:$

$\left\\{\begin{aligned}
1)\,\,\,\,(\,\partial_{a}\Phi_{1}^{(1)}\\!\\!\mid\,\,=(\,\partial_{\overline{b}}\overline{\Phi_{2}^{(1)}}\\!\\!\mid=\frac{2x\left(x^{2}-b\overline{b}\right)}{\left(x^{2}+b\overline{b}\right)^{3}},\,\,\,\,\,\,\,\,\,\,2)\,\,\,\,(\,\partial_{a}\Phi_{2}^{(1)}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\overline{\Phi_{1}^{(1)}}\\!\\!\mid=\frac{b\left(b\overline{b}-3x^{2}\right)}{\left(x^{2}+b\overline{b}\right)^{3}},\\\
3)\,\,\,\,(\,\partial_{a}\Phi_{1}^{(1)}\\!\\!\mid\,\,=(\,\partial_{b}\Phi_{2}^{(1)}\\!\\!\mid=\frac{2x\left(x^{2}-b\overline{b}\right)}{\left(x^{2}+b\overline{b}\right)^{3}},\,\,\,\,\,\,\,\,\,\,4)\,\,\,\,(\,\partial_{\overline{a}}\Phi_{2}^{(1)}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\Phi_{1}^{(1)}\\!\\!\mid=\frac{b\left(b\overline{b}-3x^{2}\right)}{\left(x^{2}+b\overline{b}\right)^{3}}.\end{aligned}\right.$

The first derivative of the function $\psi_{H}\\!\left(p\right)=p^{{-1}}$ is
also H-holomorphic at $p\in\mathbb{H}\setminus\\!\left\\{0\right\\}$ and its
left and right quaternionic derivatives become equal after the transition
$a=\overline{a}=x$ to 3D space.
The second derivative of the function $\psi_{H}\\!\left(p\right)=p^{{-1}}.$
Quite analogously, using formula (6) for $k=2$ and the calculated derivatives
$\partial_{a}\Phi_{1}^{(1)}\\!,\,\,\partial_{a}\Phi_{2}^{(1)}\\!,\,\,\partial_{\overline{a}}\Phi_{2}^{(1)}$
as well as
$\partial_{\overline{a}}\Phi_{1}^{(1)}=-\frac{2b\overline{b}\left(a+\overline{a}\right)}{\left(a\overline{a}+b\overline{b}\right)^{3}},$
we obtain after some calculations the following expression for the second
derivative of the function $\psi_{H}\\!\left(p\right)=p^{{-1}}:$

$\displaystyle\left(p^{{-1}}\right)^{{\left(2\right)}}$
$\displaystyle=\Phi_{1}^{(2)}+\Phi_{2}^{(2)}\cdot
j=\left(\partial_{a}\Phi_{1}^{(1)}\\!+\partial_{\overline{a}}\Phi_{1}^{(1)}\right)+\left(\partial_{a}\Phi_{2}^{(1)}\\!+\partial_{\overline{a}}\Phi_{2}^{(1)}\right)\cdot
j$
$\displaystyle=\frac{2\left[\overline{a}^{3}-2\overline{a}b\overline{b}-ab\overline{b}\right]+2\left[b\overline{b}-\overline{a}^{2}-a\left(\overline{a}+a\right)\right]b\cdot
j}{\left(a\overline{a}+b\overline{b}\right)^{3}},$

whence

$\Phi_{1}^{(2)}=\frac{2\left[\overline{a}^{3}-\left(2\overline{a}+a\right)b\overline{b}\right]}{\left(a\overline{a}+b\overline{b}\right)^{3}},\,\,\,\Phi_{2}^{(2)}=\frac{2\left[b\overline{b}-\left(a^{2}+a\overline{a}+\overline{a}^{2}\right)\right]b}{\left(a\overline{a}+b\overline{b}\right)^{3}}=\frac{2\left[b\overline{b}-\left(a\overline{a}\right)_{2}\right]b}{\left(a\overline{a}+b\overline{b}\right)^{3}}.$

Taking into account

$\displaystyle\overline{p}^{2}$ $\displaystyle=\left(\overline{a}-b\cdot
j\right)\cdot\left(\overline{a}-b\cdot
j\right)=\overline{a}^{2}-b\overline{b}-b\left(a+\overline{a}\right)\\!\cdot
j,$ $\displaystyle\overline{p}^{3}$
$\displaystyle=\overline{p}^{2}\cdot\overline{p}=\left[\overline{a}^{2}-b\overline{b}-b\left(a+\overline{a}\right)\\!\cdot
j\right]\cdot\left(\overline{a}-b\cdot j\right)$ $\displaystyle=$
$\displaystyle\left[\overline{a}^{3}-2\overline{a}\,b\overline{b}-ab\overline{b}\right]+\left[b\overline{b}-\overline{a}^{2}-a\left(\overline{a}+a\right)\right]\\!b\cdot
j,$ $\displaystyle p^{{-3}}=p^{{-1}}\\!\cdot p^{{-1}}\\!\cdot
p^{{-1}}=\frac{\overline{p}^{3}}{|p|^{6}}=\frac{\overline{p}^{3}}{\left(a\overline{a}+b\overline{b}\right)^{3}},$

we get the following expression for the second derivative of the function
$\psi_{H}\\!\left(p\right)=p^{{-1}}:$

$\left(p^{{-1}}\right)^{{\left(2\right)}}=2\frac{\overline{p}^{3}}{\left(a\overline{a}+b\overline{b}\right)^{3}}=2p^{{-3}}.$

We see that the obtained expression has the same form as the complex (or real)
analogue: $\left(\xi^{{-1}}\right)^{{\left(2\right)}}=2\xi^{{-3}}.$ To verify
the H-holomorphy of the second derivative
$\left(p^{{-1}}\right)^{{\left(2\right)}}$ we calculate the following
derivatives:

$\displaystyle\partial_{a}\Phi_{1}^{(2)}=\frac{2\left[-3\overline{a}^{4}+2\overline{a}\left(3\overline{a}+a\right)b\overline{b}\,-b^{2}\overline{b}^{2}\right]}{\left(a\overline{a}+b\overline{b}\right)^{4}},$
$\displaystyle\partial_{b}\Phi_{2}^{(2)}=\partial_{\overline{b}}\overline{\Phi_{2}^{(2)}}=\frac{2\left[-a\overline{a}\left(a\overline{a}\right)_{2}+2\left(a^{2}+2a\overline{a}+\overline{a}^{2}\right)b\overline{b}-b^{2}\overline{b}^{2}\right]}{\left(a\overline{a}+b\overline{b}\right)^{4}},$
$\displaystyle\partial_{a}\Phi_{2}^{(2)}=-\partial_{\overline{b}}\Phi_{1}^{(2)}=\frac{2\overline{a}\left(a^{2}+2a\overline{a}+3\overline{a}^{2}\right)b}{\left(a\overline{a}+b\overline{b}\right)^{4}}-\frac{4\left(a+2\overline{a}\right)b^{2}\overline{b}}{\left(a\overline{a}+b\overline{b}\right)^{4}},$
$\displaystyle\partial_{\overline{a}}\Phi_{2}^{(2)}=-\partial_{\overline{b}}\overline{\Phi_{1}^{(2)}}=\frac{2\left[a\overline{a}\left(\overline{a}+2a\right)b+3a^{3}b-2\left(\overline{a}+2a\right)b^{2}\overline{b}\right]}{\left(a\overline{a}+b\overline{b}\right)^{4}}.$

After performing the transition $a=\overline{a}=x$ to 3D space we see that the
H-holomorphy equations (3) are fulfilled for the second derivative of the
function $\psi_{H}\\!\left(p\right)=p^{{-1}}\\!:$

$\left\\{\begin{aligned}
&1)\,\,\,\,(\,\partial_{a}\Phi_{1}^{(2)}\\!\\!\mid\,\,=(\,\partial_{\overline{b}}\overline{\Phi_{2}^{(2)}}\\!\\!\mid=\frac{2\left(-3x^{4}+8x^{2}b\overline{b}-b^{2}\overline{b}^{2}\right)}{\left(x^{2}+b\overline{b}\right)^{4}},\\\
&2)\,\,\,\,(\,\partial_{a}\Phi_{2}^{(2)}\\!\\!\mid\,=-\,(\,\partial_{\overline{b}}\overline{\Phi_{1}^{(2)}}\\!\\!\mid=\frac{12xb\left(x^{2}-b\overline{b}\right)}{\left(x^{2}+b\overline{b}\right)^{4}},\\\
&3)\,\,\,\,(\,\partial_{a}\Phi_{1}^{(2)}\\!\\!\mid\,\,=(\,\partial_{b}\Phi_{2}^{(2)}\\!\\!\mid=\frac{2\left(-3x^{4}+8x^{2}b\overline{b}-b^{2}\overline{b}^{2}\right)}{\left(x^{2}+b\overline{b}\right)^{4}},\,\,\,\,\,\,\,\,\\\
&4)\,\,\,\,(\,\partial_{\overline{a}}\Phi_{2}^{(2)}\\!\\!\mid\,=-\,(\,\partial_{\overline{b}}\Phi_{1}^{(2)}\\!\\!\mid=\frac{12xb\left(x^{2}-b\overline{b}\right)}{\left(x^{2}+b\overline{b}\right)^{4}}.\end{aligned}\right.$

The second derivative of the H-holomorphic function
$\psi_{H}\\!\left(p\right)=p^{{-1}}$ is H-holomorphic at the points
$p\in\mathbb{H}\setminus\\!\left\\{0\right\\}$ too and its left and right
quaternionic derivatives become equal after the transition $a=\overline{a}=x$
to 3D space.

Already this example shows that calculations with quaternionic functions can
be cumbersome, error-prone and tedious manual procedures. To avoid this
problem there is a special programmes pack [13] in the Wolfram Mathematica
programming language, which verifies the commutative behavior of
multiplication of H-holomorphic functions, calculates their quaternionic
derivatives of all orders, verifies whether or not the quaternionic functions
(or their derivatives) satisfy quaternionic generalization (3) of Cauchy-
Riemann’s equations, calculates quaternionic expressions of any complexity as
well as expressions for 3D potential fields, etc. Using this programmes pack
we can easily verify that the derivatives of
$\psi_{H}\\!\left(p\right)=p^{{-1}}$ of higher orders are also all
H-holomorphic.

###  4.2 Example of natural logarithmic function

The principal value of the quaternionic natural logarithmic function:
$\psi\\!\left(p\right)=\text{Ln}\\!\left(p\right).$ The principal value of the
initial complex logarithmic function is given [12] by
$\text{Ln}\left(\xi\right)=\ln|\xi|+i\,\text{arg}\left(\xi\right)=\ln|\xi|+i\arccos\frac{\xi+\overline{\xi}}{2|\xi|}$,
where $-\pi<\text{arg}\left(\xi\right)\leq\pi$ and $|\xi|>0$. By replacing a
complex variable $\xi$ by by a quaternionic $p$ and $i$ by $r$ (as noted
above) we obtain the following expression for the principal value of the
quaternionic natural logarithmic function:

$\psi\\!\left(p\right)=\text{Ln}\left(p\right)=\ln|p|+r\cdot\text{Arccos}\frac{p+\overline{p}}{2|p|},$

where $|p|>0$ is defined by (5), $r$ by (4) and
$-\pi<\text{Arccos}\frac{p+\overline{p}}{2|p|}\leq\pi.$ Using (2), we finally
obtain the following expression for the quaternionic logarithmic function as a
function of the variables $a,\,\overline{a},\,b,\,\overline{b}:$

$\psi\\!\left(p\right)=\text{Ln}\left(p\right)=\Phi_{1}+\Phi_{2}\cdot
j=\ln|p|+\frac{\left(a-\overline{a}\right)\text{Arccos}\frac{a+\overline{a}}{2|p|}}{2v}+\frac{b\,\text{Arccos}\frac{a+\overline{a}}{2|p|}}{v}\cdot
j,$

where $v$ is the same as in (5), whence

$\Phi_{1}=\ln|p|+\frac{\left(a-\overline{a}\right)\text{Arccos}\frac{a+\overline{a}}{2|p|}}{2v},\,\,\,\Phi_{2}=\frac{b\,\text{Arccos}\frac{a+\overline{a}}{2|p|}}{v},\,\,\,p,v\neq
0.$

After the computation of the partial derivatives we have

$\displaystyle\partial_{a}\Phi_{1}=\frac{\overline{a}}{2|p|^{2}}+\frac{\theta}{2v}-\frac{\left(a-\overline{a}\right)\left[2|p|^{2}-\left(a+\overline{a}\right)\overline{a}\right]}{2|p|^{2}\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}+\frac{\left(a-\overline{a}\right)^{2}\theta}{\left(2v\right)^{3}},$
$\displaystyle\partial_{b}\Phi_{2}=\partial_{\overline{b}}\overline{\Phi_{2}}=\theta\left(\frac{1}{v}-\frac{|b|^{2}}{2v^{3}}\right)+\frac{\left(a+\overline{a}\right)|b|^{2}}{|p|^{2}\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]},$
$\displaystyle\partial_{a}\Phi_{2}=-\partial_{\overline{b}}\Phi_{1}=-\frac{b\left[2|p|^{2}-\left(a+\overline{a}\right)\overline{a}\right]}{|p|^{2}\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}+\frac{\left(a-\overline{a}\right)b\theta}{4v^{3}},$
$\displaystyle\partial_{\overline{a}}\Phi_{2}=-\partial_{\overline{b}}\overline{\Phi_{1}}=-\frac{b\left[2|p|^{2}-\left(a+\overline{a}\right)a\right]}{|p|^{2}\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}-\frac{\left(a-\overline{a}\right)b\theta}{4v^{3}},$

where $\theta=\text{Arccos}\frac{a+\overline{a}}{2|p|}$ and $a,b\neq 0.$ After
performing the transition $a=\overline{a}=x,$ we use the relations
$|p|=|p_{3}|=\sqrt{x^{2}+z^{2}+u^{2}}=\sqrt{x^{2}+|b|^{2}}>0,\,\,\,4|p|^{2}-\left(a+\overline{a}\right)^{2}=4|p_{3}|^{2}-4x^{2}=4|b|^{2}>0,\,\,\,v=\sqrt{b\overline{b}}=|b|>0,\,\,\,\theta=\text{Arccos}\frac{x}{|p_{3}|}.$
Then, substituting these relations into the above expressions for the partial
derivatives, we see that H-holomorphy equations (3) are fulfilled:

$\left\\{\begin{aligned}
1)\,\,\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\,\partial_{\overline{b}}\overline{\Phi_{2}}\\!\\!\mid=\frac{x}{2|p_{3}|^{2}}+\frac{\text{Arccos}\frac{x}{|p_{3}|}}{2|b|},\,\,\,\,\,\,\,\,\,\,2)\,\,\,(\,\partial_{a}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\overline{\Phi_{1}}\\!\\!\mid=-\frac{b}{2|p_{3}|^{2}},\\\
3)\,\,\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\,\partial_{b}\Phi_{2}\\!\\!\mid=\frac{x}{2|p_{3}|^{2}}+\frac{\text{Arccos}\frac{x}{|p_{3}|}}{2|b|},\,\,\,\,\,\,\,\,\,\,4)\,\,\,(\,\partial_{\overline{a}}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\Phi_{1}\\!\\!\mid=-\frac{b}{2|p_{3}|^{2}}.\end{aligned}\right.$

The obtained quaternionic analogue
$\psi\\!\left(p\right)=\text{Ln}\\!\left(p\right)$ of the principal value of
complex natural logarithmic function is H-holomorphic in
$\mathbb{H}\setminus\\!\left\\{0\right\\}$. The left and the right
quaternionic derivatives become equal after the transition to 3D space.
The first derivative of the function
$\psi_{H}\left(p\right)=\text{Ln}\\!\left(p\right)$. Taking into consideration
the fact that
$\partial_{\overline{a}}\Phi_{1}=\frac{a}{2|p|^{2}}-\frac{\theta}{2v}-\frac{\left(a-\overline{a}\right)^{2}\theta}{\left(2v\right)^{3}}-\frac{\left(a-\overline{a}\right)\left[2|p|^{2}-\left(a+\overline{a}\right)a\right]}{2|p|^{2}\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]},$
we get the following expression for the first derivative of the function
$\psi_{H}\\!\left(p\right)=\text{Ln}\left(p\right):$

$\displaystyle\left(\text{Ln}\left(p\right)\right)^{{\left(1\right)}}$
$\displaystyle=\Phi_{1}^{(1)}+\Phi_{2}^{(1)}\cdot
j=\left(\partial_{a}\Phi_{1}\\!+\partial_{\overline{a}}\Phi_{1}\right)+\left(\partial_{a}\Phi_{2}\\!+\partial_{\overline{a}}\Phi_{2}\right)\cdot
j$
$\displaystyle=\left\\{\frac{\overline{a}+a}{2|p|^{2}}-\frac{\left(a-\overline{a}\right)\left[2|p|^{2}-\left(a+\overline{a}\right)\overline{a}\right]}{2|p|^{2}\\!\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}-\frac{\left(a-\overline{a}\right)\left[2|p|^{2}-\left(a+\overline{a}\right)a\right]}{2|p|^{2}\\!\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}\right\\}$
$\displaystyle-\left\\{\frac{b\left[2|p|^{2}-\left(a+\overline{a}\right)\overline{a}\right]}{|p|^{2}\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}+\frac{b\left[2|p|^{2}-\left(a+\overline{a}\right)a\right]}{|p|^{2}\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}\right\\}\cdot
j$
$\displaystyle=\frac{a+\overline{a}}{2|p|^{2}}-\frac{\left(a-\overline{a}\right)\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}{2|p|^{2}\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}-\frac{b\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}{|p|^{2}\left[4|p|^{2}-\left(a+\overline{a}\right)^{2}\right]}\cdot
j$ $\displaystyle=\frac{\overline{a}}{|p|^{2}}-\frac{b}{|p|^{2}}\cdot
j=\frac{\overline{p}}{|p|^{2}}=\frac{1}{p}.$

We see that the first derivative of the quaternionic function
$\psi_{H}\\!\left(p\right)=\text{Ln}\left(p\right)$ has the same form as the
first derivative of the logarithmic function in complex (and real) analysis:
$\left(\text{Ln}\left(\xi\right)\right)^{{\left(1\right)}}\\!=\leavevmode\nobreak\
\\!\frac{1}{\xi}.$ Since the function $\frac{1}{p}$ considered in Example 4.1
above is H-holomorphic, the first derivative of the function
$\text{Ln}\left(p\right)$ is H-holomorphic too. The computation of the higher
derivatives of $\text{Ln}\left(p\right)$ repeats the results of Example 4.1.
Thus the quaternion logarithmic function
$\psi_{H}\\!\left(p\right)=\text{Ln}\left(p\right)$ possesses H-holomorphic
derivatives of all orders everywhere in
$\mathbb{H}\setminus\\!\left\\{0\right\\}\\!.$

###  4.3 Example of commutative behaviour multiplication of H-holomorphic
functions

The quaternionic multiplication of the quaternionic holomorphic functions
$\psi_{H}\\!\left(p\right)=p^{2}$ and $\psi_{H}\\!\left(p\right)=e^{p}$
behaves as commutative. Let the functions $f=f_{1}+f_{2}\cdot j$ and
$g=g_{1}+g_{2}\cdot j$ be arbitrary quaternionic functions in the
Cayley–Dickson construction. The quaternionic multiplication [10] of these
functions can be writen as follows:

$f\cdot g=\left(f_{1}+f_{2}\cdot j\right)\cdot\left(g_{1}+g_{2}\cdot
j\right)=Re\left(f\cdot g\right)+Im\left(f\cdot g\right)\cdot j,$ (8)

where

$Re\left(f\cdot
g\right)=f_{1}\,g_{1}-f_{2}\,\overline{g_{2}},\,\,\,\,\,Im\left(f\cdot
g\right)=f_{2}\,\overline{g_{1}}+f_{1}\,g_{2}$

are the designations of the parts of the quaternionic product.
The quaternionic product $p^{2}\cdot e^{p}.$ In this case we have (see
Examples 2.1, 2.2 )
$f_{1}=a^{2}-b\overline{b},\,\,f_{2}=\left(a+\overline{a}\right)b,\,\,g_{1}\\!=2\beta\cos
v+\frac{\beta\left(a-\overline{a}\right)\sin v}{v}$ and
$g_{2}=\frac{2\beta\,b\sin v}{v}.$ Subsituting these into (8) we obtain the
following expressions:

$Re\left(p^{2}\cdot
e^{p}\right)=f_{1}\,g_{1}-f_{2}\,\overline{g_{2}}=\left(a^{2}-b\overline{b}\right)\left[2\beta\cos
v+\frac{\beta\left(a-\overline{a}\right)\sin
v}{v}\right]-\left(a+\overline{a}\right)b\frac{2\beta\,\overline{b}\sin
v}{v},$ $Im\left(p^{2}\cdot
e^{p}\right)=f_{2}\,\overline{g_{1}}+f_{1}\,g_{2}=\left(a+\overline{a}\right)b\left[2\beta\cos
v+\frac{\beta\left(\overline{a}-a\right)\sin
v}{v}\right]+\left(a^{2}-b\overline{b}\right)\frac{2\beta\,b\sin v}{v},$

where we take into account (in the sequel also) that in accordance with (5) we
have $\beta=\overline{\beta},\,\,v=\overline{v}.$
The quaternionic product $e^{p}\cdot p^{2}.$ In this case we have
$f_{1}\\!=2\beta\cos v+\frac{\beta\left(a-\overline{a}\right)\sin
v}{v},\,\,f_{2}\\!=\frac{2\beta b\sin v}{v},\,\,g_{1}=a^{2}-b\overline{b}$ and
$g_{2}=\left(a+\overline{a}\right)b.$ Substituting these into (8) we obtain as
follows:

$\displaystyle Re\left(e^{p}\cdot
p^{2}\right)=f_{1}\,g_{1}-f_{2}\,\overline{g_{2}}=\left[2\beta\cos
v+\frac{\beta\left(a-\overline{a}\right)\sin
v}{v}\right]\left(a^{2}-b\overline{b}\right)-\frac{2\beta\,b\sin
v}{v}\left(a+\overline{a}\right)\overline{b},$ $\displaystyle
Im\left(e^{p}\cdot
p^{2}\right)=f_{2}\,\overline{g_{1}}+f_{1}\,g_{2}=\frac{2\beta\,b\sin
v}{v}\left(\overline{a}^{2}-b\overline{b}\right)+\left[2\beta\cos
v+\frac{\beta\left(a-\overline{a}\right)\sin
v}{v}\right]\left(a+\overline{a}\right)b.$

We see immediately that

$Re\left(p^{2}\cdot e^{p}\right)=Re\left(e^{p}\cdot p^{2}\right).$

Now we verify that $Im\left(p^{2}\cdot e^{p}\right)=Im\left(e^{p}\cdot
p^{2}\right),$ i.e.

$\displaystyle\left(a+\overline{a}\right)b\left[2\beta\cos
v+\frac{\beta\left(\overline{a}-a\right)\sin
v}{v}\right]+\left(a^{2}-b\overline{b}\right)\frac{2\beta\,b\sin v}{v}$
$\displaystyle=\frac{2\beta\,b\sin
v}{v}\left(\overline{a}^{2}-b\overline{b}\right)+\left[2\beta\cos
v+\frac{\beta\left(a-\overline{a}\right)\sin
v}{v}\right]\left(a+\overline{a}\right)b.$

Opening the brackets and simplifying the expressions, we finally obtain the
following identity:

$\left(a^{2}+\overline{a}^{2}\right)\\!\beta\sin
v=\left(a^{2}+\overline{a}^{2}\right)\\!\beta\sin v,$

that is,

$Im\left(p^{2}\cdot e^{p}\right)=Im\left(e^{p}\cdot p^{2}\right).$

Thus, it is shown that the quaternionic multiplication of the quaternionic
holomorphic functions $\psi_{H}\\!\left(p\right)=p^{2}$ and
$\psi_{H}\\!\left(p\right)=e^{p}$ behaves as commutative.

###  4.4 Example of left and right quotients equality of H-holomorphic
functions

The left quotient of $e^{p}$ by $p$ is equal to the right one. In accordance
with [14, p.20] the left $X_{L}$ and and the right $X_{R}$ quotients in this
case are correspondingly the following : $X_{L}=p^{{-1}}\\!\cdot e^{p}$ and
$X_{R}=e^{p}\cdot p^{{-1}},\,\,\,\,p\neq 0.$ Given that the functions
$p^{{-1}}$ (see Example of $p^{{-1}}$ 4.1) and $e^{p}$ (see Example of $e^{p}$
2.2) are H-holomorphic and their quaternionic multiplication behaves as
commutative, we could immediately conclude that $X_{L}=X_{R}$, however we will
show this by direct computations. Using the above expressions for $\Phi_{1}$
and $\Phi_{2}$ of the functions $p^{{-1}}$ and $e^{p},$ and calculating quite
similarly to the previous example 4.3 we get, omitting intermediate results,
the following expressions:

$\displaystyle Re\left(p^{{-1}}\\!\\!\cdot
e^{p}\right)=\frac{\overline{a}}{a\overline{a}+b\overline{b}}\left[2\beta\cos
v+\frac{\beta\left(a-\overline{a}\right)\sin
v}{v}\right]+\frac{b}{a\overline{a}+b\overline{b}}\,\frac{2\beta\,\overline{b}\sin
v}{v},$ $\displaystyle Im\left(p^{{-1}}\\!\\!\cdot
e^{p}\right)=-\frac{b}{a\overline{a}+b\overline{b}}\left[2\beta\cos
v+\frac{\beta\left(\overline{a}-a\right)\sin
v}{v}\right]+\frac{\overline{a}}{a\overline{a}+b\overline{b}}\frac{2\beta\,b\sin
v}{v},$ $\displaystyle Re\left(e^{p}\\!\\!\cdot
p^{{-1}}\right)=\left[2\beta\cos v+\frac{\beta\left(a-\overline{a}\right)\sin
v}{v}\right]\frac{\overline{a}}{a\overline{a}+b\overline{b}}+\frac{2\beta\,b\sin
v}{v}\frac{\overline{b}}{a\overline{a}+b\overline{b}}\,,$ $\displaystyle
Im\left(e^{p}\\!\\!\cdot p^{{-1}}\right)=\frac{2\beta\,b\sin
v}{v}\frac{a}{a\overline{a}+b\overline{b}}-\left[2\beta\cos
v+\frac{\beta\left(a-\overline{a}\right)\sin
v}{v}\right]\frac{b}{a\overline{a}+b\overline{b}}\,.$

We see immediately that

$Re\left(p^{{-1}}\\!\\!\cdot e^{p}\right)=Re\left(e^{p}\\!\\!\cdot
p^{{-1}}\right).$

Now we show that

$Im\left(p^{{-1}}\\!\\!\cdot e^{p}\right)=Im\left(e^{p}\\!\\!\cdot
p^{{-1}}\right)\\!,$

that is,

$\displaystyle-\frac{b}{a\overline{a}+b\overline{b}}\left[2\beta\cos
v+\frac{\beta\left(\overline{a}-a\right)\sin
v}{v}\right]+\frac{\overline{a}}{a\overline{a}+b\overline{b}}\frac{2\beta\,b\sin
v}{v}$ $\displaystyle=\frac{2\beta\,b\sin
v}{v}\frac{a}{a\overline{a}+b\overline{b}}-\left[2\beta\cos
v+\frac{\beta\left(a-\overline{a}\right)\sin
v}{v}\right]\frac{b}{a\overline{a}+b\overline{b}}\,.$

Opening the brackets and simplifying the expressions, we finally obtain the
following identity:

$\displaystyle\overline{a}\,\,\frac{\beta\,b\sin v}{v}+\frac{\beta\,ba\sin
v}{v}=\frac{\beta\,b\sin v}{v}a+\frac{\beta\,b\overline{a}\sin v}{v},$

that is,

$Im\left(p^{{-1}}\\!\\!\cdot e^{p}\right)=Im\left(e^{p}\\!\\!\cdot
p^{{-1}}\right).$

We have shown that the left quotient of $e^{p}$ by $p$ is equal to the right
one and at the same time confirmed once again that the quaternionic
multiplication of H-holomorphic functions behaves as commutative.

The commutativity property of quternionic multiplication of H-holomorphic
functions as well as the equality of the left and the right quotients of
H-holomorphic functions can be illustrated with special computer programs [13]
when using various examples of H-holomorphic functions

As shown above, quaternionic generalization (3) of complex Cauchy-Riemann’s
equations is based on the requirement of uniqueness of a quaternionic
derivative, which is needed to explore steady state vector fields in 3D space.
It is not superfluous to note that the physical formulation of a problem
played initially an important role in the theory of complex-differentiable
functions, and the so-called complex Cauchy-Riemann equations were found [17]
as early as in 1752 in d’Alembert’s doctrine about planar fluid flow.

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