# On Decompositions of H-holomorphic
functions into quaternionic power series

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

###### Abstract

Based on the full similarity in algebraic properties and differentiation rules
between quaternionic (H-) holomorphic and complex (C-) holomorphic functions,
we assume that there exists one holistic notion of a holomorphic function that
has a H-representation in the case of quaternions and a C-representation in
the case of complex variables. We get the essential definitions and criteria
for a quaternionic power series convergence, adapting complex analogues to the
quaternion case. It is established that the power series expansions of any
holomorphic function in C- and H-representations are similar and converge with
identical convergence radiuses. We define a H-analytic function and prove that
every H-holomorphic function is H-analytic. Some examples of power series
expansions are given.

2020 Mathematics Subject Classification: 30G35.
Keywords: quaternionic analysis, quaternionic Cauchy-Riemann’s equations,
quaternionic holomorphic functions, quaternionic analytic functions,
decomposition into quaternionic power series
Date: July 31, 2024
e-mail address<EMAIL_ADDRESS>as well as<EMAIL_ADDRESS>
##  1 Introduction

A possibility of a decomposition of functions into power series is interesting
in particular because it defines the notion of analytic functions in classical
mathematical analysis. For spaces of dimension more than two, for example,
there are works devoted to quaternionic power series in the cases of regular
[1] and slice-regular [2] quaternionic functions as well as in the more
generalized case of Clifford-holomorphic [3] functions. We discuss here some
aspects of power series expansions within the framework of the theory of the
so-called H-holomorphic functions [4], [5], uniting the left and the right
approaches. Now, we recall some results from this theory [4] needed for the
sequel.

An independent quaternionic vaviable $p$ is further denoted by

$p=x+yi+zj+uk,$

and correspondingly a quaternion-valued function $\psi\left(p\right)$ by

$\psi\left(p\right)=\psi_{1}\left(x,y,z,u\right)+\psi_{2}\left(x,y,z,u\right)i+\psi_{3}\left(x,y,z,u\right)j+\psi_{4}\left(x,y,z,u\right)k,$

where $x,y,z,u$ and
$\psi_{1}\left(x,y,z,u\right),\psi_{2}\left(x,y,z,u\right),\psi_{3}\left(x,y,z,u\right),\psi_{4}\left(x,y,z,u\right)$
are real-valued, and $i,j,k$ are the base quaternions of the quaternion space
$\mathbb{H}$ defined by the expressions

$i^{2}=j^{2}=k^{2}=-1,$ $ij=-ji=k,\,\,\,\,jk=-kj=i,\,\,\,\,ki=-ik=j.$

In the Cayley–Dickson construction (doubling form, doubling procedure) [6, p.
41] 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 also use 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 quaternionic
imaginary unit $j$ in the Cayley–Dickson doubling form should always be
recorded behind the complex function
$\Phi_{2}\left(a,b,\overline{a},\overline{b}\right)$, to achieve this goal we
often use the known [6, p. 42] relation $j\alpha=\overline{\alpha}j$,
where $\alpha$ is any complex function.

Further, we use the following necessary and sufficient conditions ([4, p.
15], [5, p. 4]) for a function $\psi\left(p\right)$ to be H-holomorphic:

######  Definition 1.1.

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 $\psi\left(p\right)=\Phi_{1}+\Phi_{2}\cdot j$ possess continuous
first-order partial derivatives with respect to $a,\overline{a},b,$ 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 (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 Cauchy-Riemann’s equations:

$\left\\{\begin{aligned}
1)\,\,\,\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\,\partial_{\overline{b}}\overline{\Phi_{2}}\\!\mid,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2)\,\,\,\,(\,\partial_{a}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\overline{\Phi_{1}}\\!\mid,\\\
3)\,\,\,\,(\,\partial_{a}\Phi_{1}\\!\\!\mid\,\,=(\,\partial_{b}\Phi_{2}\\!\mid,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4)\,\,\,\,(\,\partial_{\overline{a}}\Phi_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\Phi_{1}\\!\\!\mid.\end{aligned}\right.$
(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: $p=x+yi+zj+uk\rightarrow p_{3}=x+zj+uk$) has been already
performed in expressions enclosed in brackets. Equations (3-1) and (3-2) have
the components relating to the left quaternionic derivative [5, p. 4] and
equations (3-3) and (3-4) to the right one. The requirement $a=\overline{a}=x$
provides a 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 [5, p. 4], i.e. the left derivative becomes equal
to the right one.

The 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. Nevertheless, this doesn’t mean that we deal with triplets,
since the transition $a=\overline{a}=x$ (or $y=0$) can not be initially done
for quaternionic variables and functions. Otherwisewe we would lose a division
operation. 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. More clearly, they are those quaternionic functions whose the left and
the right derivatives become equal after the transition to 3D space.

Denoting the complex space (plane) by $\mathbb{C}$ and a complex holomorphic
(C-holomorph-
ic) function by $\psi_{C}\\!\left(\xi\right)$, where $\xi$ is an independent
complex variable, we have the following

######  Theorem 1.2.

Let a complex-valued function
$\psi_{C}\\!\left(\xi\right):G_{2}\rightarrow\mathbb{C}$ be C-holomorphic
everywhere in an open connected set $G_{2}\subseteq\mathbb{C}$, except,
possibly, at certain singularities. Then a H-holomorphic function
$\psi_{H}\\!\left(p\right)$ of the same kind as $\psi_{C}\\!\left(\xi\right)$
can be constructed (without change of a kind of function) from
$\psi_{C}\\!\left(\xi\right)$ by replacing a complex variable $\xi\in G_{2}$
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 transition from $p$ to $\xi$.

We also need the following [4, p. 15]

######  Theorem 1.3.

Let a continuous quaternion-valued function
$\psi_{H}\\!\left(p\right)=\Phi_{1}+\Phi_{2}\cdot j,$ where
$\Phi_{1}\left(a,b,\overline{a},\overline{b}\right)$ and
$\Phi_{2}\left(a,b,\overline{a},\overline{b}\right)$ are differentiable with
respect to $a,b,\overline{a}$ and $\overline{b}$, be H-holomorphic everywhere
in its domain of definition $G_{4}\subseteq\mathbb{H}$. Then its full
quaternionic derivative (uniting the left and the right derivatives) with
respect to $p$ defined by

$\psi_{H}^{{}^{\prime}}\\!\\!\left(p\right)=\Phi_{1}^{{}^{\prime}}+\Phi_{2}^{{}^{\prime}}\cdot
j,$ (4)

where

$\Phi_{1}^{{}^{\prime}}=\partial_{a}\Phi_{1}+\partial_{\overline{a}}\Phi_{1},\,\,\,\,\Phi_{2}^{{}^{\prime}}=\partial_{a}\Phi_{2}+\partial_{\overline{a}}\Phi_{2},$
(5)

is also H-holomorphic in $G_{4}.$ If a quaternion function
$\psi_{H}\\!\left(p\right)$ is once quaternion-
differentiable with respect to $p$ in $G_{4},$ then it possesses the full
quaternionic derivatives of all orders in $G_{4},$ each one H-holomorphic.

The generalized formula for the full $k^{\prime}th$ quaternionic derivative
[5, p. 7] is the following:

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

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

$\Phi_{1}^{(k)}=\partial_{a}\Phi_{1}^{(k-1)}\\!\\!+\partial_{\overline{a}}\Phi_{1}^{(k-1)},\,\,\,\,\,\,\,\Phi_{2}^{(k)}=\partial_{a}\Phi_{2}^{(k-1)}\\!\\!+\partial_{\overline{a}}\Phi_{2}^{(k-1)}$
(7)

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
$\psi_{H}^{\left(k-1\right)}\\!\\!\left(p\right)=\Phi_{1}^{(k-1)}+\Phi_{2}^{(k-1)}\cdot
j,k\geq 1;$
$\Phi_{1}^{(0)}=\Phi_{1}\\!\\!\left(a,b,\overline{a},\overline{b}\right)\text{and}\,\,\Phi_{2}^{(0)}=\Phi_{2}\\!\\!\left(a,b,\overline{a},\overline{b}\right)\\!.$

Theorem 1.3 leads [4, p. 15] to the following

######  Corollary 1.4.

(a similarity between differentiating rules for C-holomorhic and H-holomorhic
functions). All expressions for derivatives of a H-holomorphic function
$\psi_{H}\\!\left(p\right)$ of the same kind as a C-holomorphic analogue
$\psi_{C}\\!\left(\xi\right)$ have the same forms as the expressions for
corresponding derivatives of a function $\psi_{C}\\!\left(\xi\right).$

For example, if the second derivative of the C-holomorphic function
$\psi_{C}\\!\left(\xi\right)=\xi^{k},\,\,k=1,2,3,\dots,$ is
$\psi_{C}^{\left(2\right)}\\!\left(\xi\right)=k\left(k-1\right)\xi^{k-2},$
then the second full derivative of the H-holomorphic function
$\psi_{H}\\!\left(p\right)=p^{k}$ is
$\psi_{H}^{\left(2\right)}\\!\left(p\right)=k\left(k-1\right)p^{k-2}.$
Corollary 1.4 is illustrated by numerous examples of elementary quaternionic
functions presented in [4], [5], [7].

Such a similarity is based on the above-mentioned uniting (see [7, p. 18])
the left and the right quaternionic derivatives, giving the full quaternionic
derivative (4). Indeed, the component $\Phi_{2}^{{}^{\prime}}$ in (5) consists
of the derivatives $\partial_{a}\Phi_{2}$ and
$\partial_{\overline{a}}\Phi_{2}.$ The derivative $\partial_{a}\Phi_{2}$
belongs to equation (3-2) associated with the left quaternionic derivative,
and the derivative $\partial_{\overline{a}}\Phi_{2}$ belongs to equation (3-4)
associated with the right quaternionic derivative. This also applies to higher
derivatives.

The goal of this article is to show that the full similarity in algebraic
properties and differentiation rules between the C- and H-holomorphic
functions, based on Theorems 1.2, 1.3, and Corollary 1.4 , can be extended to
decompositions of H-holomorphic functions into infinite quaternionic power
series.

##  2 H-holomorphic linear combinations

We need to establish, under what conditions a finite linear combination of
H-holomorphic functions can also be H-holomorphic. We have first of all to
consider the following

######  Proposition 2.1.

(multiplying by a constant). A H-holomorphic function $f_{H}\\!\left(p\right)$
multiplied by a constant $r$:

$\psi_{H}\\!\left(p\right)=rf_{H}\\!\left(p\right)$ (8)

is also H-holomorphic only if a constant $r$ is real-valued. Then the full
$k^{\prime}th$ order H-holomorphic derivative of a H-holomorphic function
multiplied by a constant is the following:

$\psi_{H}^{(k)}\\!\left(p\right)=\left(rf_{H}\\!\left(p\right)\right)^{\left(k\right)}=rf_{H}^{\left(k\right)}\\!\left(p\right).$
(9)

###### Proof.

Let a quaternionic function
$f_{H}\\!\left(p\right)=f_{1}\\!\left(a,b,\overline{a},\overline{b}\right)+f_{2\cdot}\\!\left(a,b,\overline{a},\overline{b}\right)\cdot
j$ be H-holomorphic. Then it satisfies the conditions of H-holomorphy (3) as
follows:

$\left\\{\begin{aligned}
1)\,\,\,\,(\,\partial_{a}f_{1}\\!\\!\mid\,\,=(\,\partial_{\overline{b}}\overline{f_{2}}\\!\mid,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2)\,\,\,\,(\,\partial_{a}f_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}\overline{f_{1}}\\!\mid,\\\
3)\,\,\,\,(\,\partial_{a}f_{1}\\!\\!\mid\,\,=(\,\partial_{b}f_{2}\\!\mid,\,\,\,\,\,\,\,\,\,\,\,\,\,\,4)\,\,\,\,(\,\partial_{\overline{a}}f_{2}\\!\\!\mid&=-\,(\,\partial_{\overline{b}}f_{1}\\!\\!\mid.\end{aligned}\right.$
(10)

As it has been found in [7, p. 19], there exist in addition to main
equations (3) for the H-holomorphic functions
$\psi_{H}\left(p\right)=\Phi_{1}+\Phi_{2}\cdot j$ also the following
equations:

$\left\\{\begin{aligned}
a)\,\,\,\,\,\partial_{b}\Phi_{2}\\!&=\,\partial_{\overline{b}}\overline{\Phi_{2}},\\!\,\,\,\,\,\,\,\,\,\,\,\,\,\,b)\,\,\,\,\partial_{a}\Phi_{2}\\!\\!\,\,=-\,\partial_{\overline{b}}\Phi_{1},\\!\\\
c)\,\,\,\,\,\partial_{\overline{a}}\Phi_{1}\\!&=\,\,\partial_{a}\overline{\Phi_{1}},\\!\,\,\,\,\,\,\,\,\,\,\,\,\,d)\,\,\,\,\partial_{\overline{a}}\Phi_{2}=-\,\,\partial_{\overline{b}}\overline{\Phi_{1}},\end{aligned}\right.$
(11)

which hold true but already without having to carry out the transition to 3D
space. These equations are illustrated by all presented in [4], [5] examples
of H-holomorphic functions.

Consider a quaternionic product $\psi\left(p\right)=C\cdot
f_{H}\\!\left(p\right)\\!,$ where we initially suppose that the constant
$C=C_{1}+C_{2}\cdot j$ is quaternion-valued, i.e. $C_{1}$ and $C_{2}$ are
complex-valued constants. We need to define values of constants $C_{1}$ and
$C_{2}$ for which the function $\psi\left(p\right)=C\cdot
f_{H}\\!\left(p\right)=\Phi_{1}+\Phi_{2}\cdot j$ is H-holomorphic. Let us
recall the multiplication rule for quaternions in the Cayley–Dickson doubling
form [6, p. 42] :

$p\cdot q=\left(a_{1}+b_{1}\cdot j\right)\cdot\left(a_{2}+b_{2}\cdot
j\right)=\left(a_{1}a_{2}-b_{1}\overline{b_{2}}\right)+\left(a_{1}b_{2}+\overline{a_{2}}\,b_{1}\right)\cdot
j,$ (12)

where

$\displaystyle p=a_{1}+b_{1}\cdot j,\,\,\,\,\,\,\,\,q=a_{2}+b_{2}\cdot j,$
$\displaystyle a_{1}=x_{1}+y_{1}i,\,\,\,\,\,\,\,\,\,\,b_{1}=z_{1}+u_{1}i,$
$\displaystyle a_{2}=x_{2}+y_{2}i,\,\,\,\,\,\,\,\,\,\,b_{2}=z_{2}+u_{2}i.$

Using this rule, we obtain as follows:

$\psi\left(p\right)=C\cdot f_{H}\\!\left(p\right)=\left(C_{1}+C_{2}\cdot
j\right)\cdot\left(f_{1}+f_{2}\cdot
j\right)=\left(C_{1}f_{1}-C_{2}\overline{f_{2}}\right)+\left(C_{1}f_{2}+C_{2}\overline{f_{1}}\right)\cdot
j,$

whence

$\Phi_{1}=C_{1}f_{1}-C_{2}\overline{f_{2}},\,\,\,\,\,\,\,\Phi_{2}=C_{1}f_{2}+C_{2}\overline{f_{1}}$
(13)

and correspondingly

$\overline{\Phi_{1}}=\overline{C_{1}}\,\overline{f_{1}}-\overline{C_{2}}f_{2},\,\,\,\,\,\,\,\overline{\Phi_{2}}=\overline{C_{1}}\,\overline{f_{2}}+\overline{C_{2}}f_{1}.$
(14)

To prove that a constant can be only a real value, it suffices to use (11 -a).
Differentiating $\Phi_{2}$ with respect to $b$ and $\overline{\Phi_{2}}$ with
respect to $\overline{b},$ and substituting the obtained derivatives into
(11-a) we get the following equation:

$\partial_{b}\Phi_{2}=C_{1}\partial_{b}f_{2}+C_{2}\partial_{b}\overline{f_{1}}=\partial_{\overline{b}}\overline{\Phi_{2}}=\overline{C_{1}}\partial_{\overline{b}}\overline{f_{2}}+\overline{C_{2}}\partial_{\overline{b}}f_{1},$

that must be satisfied if the function $\psi\left(p\right)=C\cdot
f_{H}\\!\left(p\right)=\Phi_{1}+\Phi_{2}\cdot j$ must be H-holomorphic. Since
$f_{H}\\!\left(p\right)$ is H-holomorphic, the equality
$\partial_{b}f_{2}=\partial_{\overline{b}}\overline{f_{2}}$ holds true in this
condition, and hence the equality
$C_{1}\partial_{b}f_{2}=\overline{C_{1}}\partial_{\overline{b}}\overline{f_{2}}$
can only be satisfied if $C_{1}=\overline{C_{1}}.$

The derivative $\partial_{b}\overline{f_{1}}$ does not belong to equations
(10) when using them for the H-holomorphic function
$f_{H}\\!\left(p\right))=f_{1}+f_{2}\cdot j$, hence the condition
$\partial_{b}\overline{f_{1}}=\partial_{\overline{b}}f_{1}$ cannot be in
general satisfied. In this case the equality
$C_{2}\partial_{b}\overline{f_{1}}=\overline{C_{2}}\partial_{\overline{b}}f_{1}$
can only be satisfied if $C_{2}=\overline{C_{2}}=0.$

It is not difficult to show that the function $\psi\left(p\right)=C\cdot
f_{H}\\!\left(p\right)$ satisfies all equations of system (3), if
$C_{1}=\overline{C_{1}}$ and $C_{2}=0.$ For example, consider equation (3-2).
If this equation must be satisfied, then, differentiating the functions
$\Phi_{2}$ and $\overline{\Phi_{1}}$ defined, respectively, by (13) and (14),
we get the following equation:

$(\partial_{a}\Phi_{2}\\!\\!\mid=C_{1}(\partial_{a}f_{2}\\!\\!\mid+C_{2}(\partial_{a}\overline{f_{1}}\\!\\!\mid=-(\partial_{\overline{b}}\overline{\Phi_{1}}\\!\\!\mid=-\overline{C_{1}}(\partial_{\overline{b}}\overline{f_{1}}\\!\\!\mid+\overline{C_{2}}(\partial_{\overline{b}}f_{2}\\!\\!\mid,$

which must be satisfied as well. Since, in accordance with equation (10-2), we
have
$(\partial_{a}f_{2}\\!\\!\mid=-(\partial_{\overline{b}}\overline{f_{1}}\\!\\!\mid,$
the partial equality
$C_{1}(\partial_{a}f_{2}\\!\\!\mid=-\overline{C_{1}}(\partial_{\overline{b}}\overline{f_{1}}\\!\\!\mid$
will be satisfied only if $C_{1}=\overline{C_{1}}.$ On the other hand, since
the function $(\partial_{a}\overline{f_{1}}\\!\\!\mid$ doesn’t belong to
system (10), the equality
$(\partial_{a}\overline{f_{1}}\\!\\!\mid=(\partial_{\overline{b}}f_{2}\\!\\!\mid$
cannot be satisfied when using system (10). We conclude that the equality
$C_{2}(\partial_{a}\overline{f_{1}}\\!\\!\\!\mid=\overline{C_{2}}(\partial_{\overline{b}}f_{2}\\!\\!\\!\mid$
can be satisfied only if $C_{2}=0.$

Thus, the function $\psi\left(p\right)=C\cdot f_{H}\\!\left(p\right)$
satisfies equation (3-2) only if
$C_{1}=\overline{C_{1}}=r\in\leavevmode\nobreak\ \mathbb{R}$ and $C_{2}=0,$
i.e. $C=r\in\mathbb{R}.$ Quite analogously we can verify the validity of the
remaining H-holomorphy equations of system (3) if
$C_{1}=\overline{C_{1}}=r\in\mathbb{R}$ and $C_{2}=0.$ The same results we
obtain for H-holomorphic functions originally multiplied on the right by a
constant $C.$

We finally conclude that in order to get a H-holomorphic product of a
H-holomorphic function by a constant, the constant can be only real-valued.
Using Corollary 1.4, further, we prove expression (9). ∎

######  Proposition 2.2.

(H-holomorphic linear combinations) A sum of a finite number $n$ of
H-holomorphic functions $f_{l}\\!\left(p\right)$ each multiplied by a real
constant $r_{l}:$

$\psi_{H}\\!\left(p\right)=\sum_{l=0}^{n}r_{l}\,f_{l}\\!\left(p\right)\\!,$
(15)

where
$\mid\\!r_{1}\\!\\!\mid^{2}+\mid\\!r_{2}\\!\\!\mid^{2}+\cdots+\mid\\!r_{n}\mid^{2}\neq
0,$ is H-holomorphic as well. The full $k^{\prime}th$ order H-holomorphic
derivative of $\psi_{H}\\!\left(p\right)$ is the following:

$\psi_{H}^{(k)}\\!\left(p\right)=\left(\sum_{l=0}^{n}r_{l}\,f_{l}\\!\left(p\right)\right)^{\\!\\!\\!\left(k\right)}\\!=\sum_{l=0}^{n}r_{l}\,f_{l}^{\left(k\right)}\left(p\right).$

###### Proof.

It follows from Proposition 2.1, linearity of equations (3) and Corollary 1.4
. ∎

##  3 Power series expansions of H-holomorphic functions

The algebraic properties of addition, subtraction, multiplication, and
division of H-holomorphic functions are fully identical (isomorphic) with
complex ones. This is based on the established facts that the quaternionic
multiplication of the H-holomorphic functions behaves as commutative (see the
proof in [4, p. 18]) and the left quotient of them is equal to the right
one (see the proof in [4, p.20]). As a result of algebraic operations with
H-holomorphic functions we also get H-holomorphic functions [4, p. 17].

It is evident that due to these facts, the commutative and associative laws
for addition and multiplication, and the distributive laws in the case of the
H-holomorphic functions are the sames as in the case of C-holomorphic
functions. Note, however, that this only applies to holomorphic functions
$\psi_{H}\\!\left(p\right)$ and $\psi_{C}\\!\left(\xi\right)$. Thus, one can
speak of the isomorphism between the algebras of H-holomorphic and
C-holomorphic functions. The differentiation rules for H-holomorphic functions
are also fully similar to complex ones [4, p. 17] and results of
differentiating are also H-holomorphic.

Formula (12) for quaternionic mulniplication gives as a particular case the
formula for complex multiplication (see [4, p. 18]) after the transition to
the complex case, i.e. at
$a_{1}=x_{1}\left(y_{1}=0\right),b_{1}=z_{1}\left(u_{1}=0\right),a_{2}=\overline{a_{2}}=x_{2}\left(y_{2}=0\right),b_{2}=\overline{b_{2}}=z_{2}\left(u_{2}=0\right)$.
Quite analogously we get the complex Cauchy-Riemann equations as a particular
case [5, p. 8] of the necessary and sufficient conditions (3) for a
quaternionic function to be H-holomorphic. Given these facts, we can formulate
the following

######  Proposition 3.1.

We assume that there exists in principle one holistic notion of a holomorphic
function that in the case of quaternions is represented by a H-holomorphic
function and in the case of complex variables by a C-holomorphic function.
Replacement of a complex variable $\xi$ by a quaternionic $p$ carries out in
accordance with Theorem 1.2 the transition from the complex representation of
a holomorphic function in a set $G_{2}$ to its quaternionic representation in
a set $G_{4}$. Correspondingly, we can speak of a C-representation and a
H-representation of one holistic notion of holomorphic functions.

According to Theorem 1.2, the quaternionic power functions
$p^{l},\,l=0,1,2,\dots,$ and their derivatives of all orders are
H-holomorphic. Putting $f_{l}\left(p\right)=p^{l}$ in (15), we obtain the
following H-holomorphic polynomial:

$\psi_{H}\left(p\right)=S_{n}\left(p\right)=\sum_{l=0}^{n}r_{l}p^{l},\,n=1,2,3,\dots,\,\,\,\,\,\,r_{l}\in\mathbb{R},\,p\in\mathbb{H}.$
(16)

It is natural to introduce an object that is like a H-holomorphic polynomial,
but with infinitely many terms. Continuing (16) by increasing the $n,$ we get
the following quaternionic power series:

$\psi\left(p\right)=S\left(p\right)=\sum_{l=0}^{\infty}r_{l}p^{l}=r_{0}+r_{1}p^{1}+r_{2}p^{2}+\cdots+r_{l}p^{l}+\cdots,\,\,\,\,\,\,r_{l}\in\mathbb{R},\,p\in\mathbb{H},$
(17)

which is a Taylor series $\sum_{l=0}^{\infty}r_{l}\left(p-p_{0}\right)^{l}$ at
$p_{0}=0$ or a Maclaurin series [10, p. 353] for the function
$\psi\left(p\right)$.

Since all algebraic operations and differentiating rules for H-holomorphic
power functions are identical with those in complex analysis, we can introduce
notions related to quaternionic power series similarly to complex ones. Given
Proposition 3.1, it is not unreasonable to expect that if a C-representation
of a holomorphic function by a convergent power series exists, then a
H-representation of this holomorphic function by a convergent power series
must also exist. However, we would like to verify this directly, and therefore
we need first to adapt some complex definitions and convergence criteria to
the quaternionic case. This is also important because we want to show a
consistency between complex and quaternionic power series.

As in complex analysis [10], a notion of an infinite quaternionic series
$\sum_{l=0}^{\infty}r_{l}p^{l}$ is based on a notion of a convergent sequence
$\left(S_{n}\left(p\right)\right)$ of partial sums
$S_{n}\left(p\right)=\sum_{k=0}^{n}r_{k}p^{k},\\\ n=0,1,2,\dots$.

By analogy with [8, p. 99] we introduce the following

######  Definition 3.2.

An infinite quaternionic series $\sum_{l=0}^{\infty}r_{l}p^{l}$ is said to be
convergent if the sequence $\left(S_{n}\left(p\right)\right)$ of the partial
sums $S_{n}\left(p\right)=\sum_{l=0}^{n}r_{l}\,p^{l}$ is convergent. At that,
$\lim_{n\rightarrow{\infty}}S_{n}\left(p\right)=S\left(p\right)$ is said to be
a sum $S\left(p\right)$ of this infinite series and $r_{l}p^{l}$ its term.
Otherwise the series is divergent. A power series
$\sum_{l=0}^{\infty}r_{l}p^{l}$ is said to be absolutely convergent if a
series
$\sum_{l=0}^{\infty}\mid\\!\\!r_{l}\,p^{l}\\!\\!\mid\,=\\!\sum_{l=0}^{\infty}\\!r_{l}\\!\mid\\!\\!p\\!\\!\mid^{l},$
where
$\mid\\!\\!p\\!\\!\mid=\sqrt{x^{2}+y^{2}+z^{2}+u^{2}}=\sqrt{a\overline{a}+b\overline{b}},$
converges. A power series $\sum_{l=0}^{\infty}r_{l}p^{l}$ is said to be
conditionally convergent if it converges but not absolutely.

Analogously to complex analysis [8, p. 209] we also introduce the following

######  Definition 3.3.

A quaternionic series $\sum_{l=0}^{\infty}r_{l}p^{l}$ is said to be uniformly
convergent to its sum $S\left(p\right)$ on the set $T\\!\subset\mathbb{H}$ if
it converges at all $p\in T$ to $S\left(p\right)$, and for every $\epsilon>0$
there exists an integer $N_{\epsilon}>0$ (that depends only on $\epsilon$)
such that if $n\geq N_{\epsilon}$, then $\mid
S_{n}\left(p\right)-S\left(p\right)\mid<\epsilon$ for all $p\in T.$

We generalize the complex definition of Cauchy criterion [10, p. 250] to
the quaternionic case as follows.

######  Definition 3.4.

(a quaternionic Cauchy criterion) A quaternionic series
$\sum_{l=0}^{\infty}r_{l}\,p^{l}$ converges uniformly to $S\left(p\right)$ on
$T$ if and only if for every $\epsilon>0$ there is an integer $N_{\epsilon}>0$
such that $\mid\\!S_{n}\left(p\right)-S_{m}\left(p\right)\\!\mid<\epsilon$ for
all $n,m>N_{\epsilon}$ and for all $p\in T.$

We associate every quaternionic power series with a real number $R\geq 0$
called a radius of convergence of a series [8, p. 110]. We further define a
quaternionic generalization of a complex domain of convergence as an open
connected ball
$B\left(0,R\right)=\left\\{p\\!:\mid\\!p\mid<R\right\\}\subset\mathbb{H}$
centered at $p=0$ such that
$\psi\left(p\right):B\left(0,R\right)\rightarrow\mathbb{H},$ where $R$ is a
radius of convergrnce. For simplicity, we do not consider functions with
singularities.

Further, we reproduce some assertions and theorems from complex analysis [8],
[9], [10] as “propositions” adapted to quaternions. At that, we repeat complex
proofs, adapting them to the quaternionic case. We begin by the following
important property of convergent series.

######  Proposition 3.5.

(a term test). The only series $\sum_{l=0}^{\infty}r_{l}\,p^{l}$ that can
converge are those ones whose terms approach $0.$ In other words, if
$\sum_{l=0}^{\infty}r_{l}\,p^{l}$ converges, then $r_{l}\,p^{l}\rightarrow 0$
as $l\rightarrow{\infty}.$

###### Proof.

If the series $\sum_{l=0}^{\infty}r_{l}\,p^{l}$ converges in
$B\left(0,R\right)$, then the limit of the sequence of its partial sums
approaches the sum $S$, i.e. $S_{n}\rightarrow S$ as $n\rightarrow{\infty}$,
where $S_{n}$ is the $n^{\prime}th$ partial sum
$S_{n}=\sum_{k=0}^{n}r_{k}\,p^{k}$. Putting $n=l$, we have

$\lim_{l\rightarrow\infty}r_{l}\,p^{l}=\lim_{l\rightarrow\infty}\left(S_{l}-S_{l-1}\right)=\lim_{l\rightarrow\infty}S_{l}-\lim_{l\rightarrow\infty}S_{l-1}=S-S=0.$

∎

The contrapositive of that statement gives a test which can tell us that some
series diverge. Note, however, the terms converging to $0$ don’t imply the
series converges. The term test is only necessary, but not a sufficient
convergence test.

Analogously to [8, p. 105] we introduce the following

######  Proposition 3.6.

(d’Alembert’s Ratio Test). If the series $\sum_{l=0}^{\infty}r_{l}\,p^{l}$ is
a quaternionic power series with the property that

$\lim_{l\rightarrow\infty}\frac{\mid\\!r_{\left(l+1\right)}\,p^{{l+1}}\\!\mid}{\mid\\!r_{l}\,p^{l}\\!\mid}=L$
(18)

(provided the limit exists), then the series is absolutely convergent in an
open connected ball $B\left(0,\frac{1}{L}\right)$ if $L<1$ and divergent if
$L>1$. If $L=1$, then the test is inconclusive, so we have to use some other
test. If the series is convergent, then $R=\frac{1}{L}$ (possibly infinite) is
a radius of convergence.

###### Proof.

Since we only deal with positive values in (18), the proof of this proposition
is quite analogous to complex one [12, p. 193]. We skip the details. ∎

The Weierstrass M-test, given in [8, p. 210], can be generalized to the
quaternionic case as follows.

######  Proposition 3.7.

(a quaternionic Weierstrass M-test). Suppose the infinite quaternionic series
$\sum_{l=0}^{\infty}r_{l}\,p^{l}$ has the property

$\mid\\!\\!r_{l}\,p^{l}\\!\\!\mid\leq M_{l},\,\,\,M_{l}\geq 0$

for each $l\in\left\\{0,1,2,\dots\right\\}$ and all $p\in T.$ If the series
$\sum_{l=0}^{\infty}M_{l}$ converges, then the series
$\sum_{l=0}^{\infty}r_{l}\,p^{l}$ converges uniformly and absolutely on $T.$

###### Proof.

Consider the sequence $\left\\{S_{n}\left(p\right)\right\\}$ of partial sums
$S_{n}\left(p\right)=\sum_{l=0}^{n}r_{l}\,p^{l}$. If $n>m,$ where
$n,m\in\mathbb{N}$, then $\mid\leavevmode\nobreak\
\\!\\!\\!S_{n}\left(p\right)-S_{m}\left(p\right)\\!\\!\mid=\mid\\!\\!\\!\\!\\!\leavevmode\nobreak\
\\!\sum_{l=m+1}^{n}r_{l}\,p^{l}\\!\\!\mid=\sum_{l=m+1}^{n}\\!\\!\mid\\!\\!r_{l}\,p^{l}\\!\\!\mid\leq\sum_{l=m+1}^{n}M_{l}.$
Since the series $\sum_{l=0}^{\infty}M_{l}$ converges, the last expression can
be made as small as we wish by choosing the positive integer $m$ large enough.
Thus for a given value $\epsilon>0$ there is a positive integer $N_{\epsilon}$
such that if $n,m>N_{\epsilon},$ then $\mid\leavevmode\nobreak\
\\!\\!\\!S_{n}\left(p\right)-S_{m}\left(p\right)\\!\\!\mid<\epsilon.$ But this
means that for all $p\in T,$ according to Definition 3.4, the series
$\sum_{l=0}^{\infty}r_{l}\,p^{l}$ converges uniformly on a set $T$. Since the
series $\sum_{l=0}^{\infty}\mid\\!r_{l}\,p^{l}\\!\mid$ is majorized by the
convergent series $\sum_{l=0}^{\infty}M_{l}$, the series
$\sum_{l=0}^{\infty}r_{l}\,p^{l}$ also converges absolutely on $T$. ∎

######  Proposition 3.8.

(terms bounded by a constant) If terms of a quaternionic power series
$\sum_{l=0}^{\infty}r_{l}\,p^{l}$ are bounded at some point
$p_{0}\in\mathbb{H}$:

$\mid\\!r_{l}p_{0}^{l}\mid\leq M,$ (19)

where $M>0$ is a constant and $l=0,1,2,\dots$, then this series converges in
an open connected ball
$B\left(0,\mid\\!p_{0}\\!\mid\right)=\left\\{p:\,\mid\\!p\\!\mid<\mid
p_{0}\\!\mid\right\\}$. Moreover, it converges absolutely and uniformly on any
compact subset $K$ that is properly contained in
$B\left(0,\mid\\!p_{0}\mid\right)$.

###### Proof.

We adapt the proof considered in [9, p. 96] to the quaternionic case.
Suppose that $p_{0}\neq 0$ and  $\mid\\!p_{0}\\!\\!\mid=\zeta>0$. Otherwise
the ball $B\left(0,\zeta\right)$ is empty. Let $K\in
B\left(0,\zeta\right)=\left\\{p:\,\mid\\!p\mid<\zeta\right\\}$, then there
exists $q<1$ such that $\frac{\mid p\mid}{\zeta}\leq q<1$ for all $p\in K$.
Therefore for any $p\in K$ and any of the above $l$ we have
$r_{l}\\!\\!\mid\\!p^{l}\mid\leq r_{l}\,\zeta^{l}q^{l}$. However, inequality
(19) implies that $\mid\\!r_{l}p_{0}^{l}\mid=r_{l}\zeta^{l}\leq M$, then
$r_{l}\\!\\!\mid\\!p^{l}\mid\leq Mq^{l}$ and the series
$\sum_{l=0}^{\infty}r_{l}\,p^{l}$ is majorized by the convergent series
$M\sum_{l=0}^{\infty}q^{l}$ (the geometric series
$\sum_{l=0}^{\infty}q^{l}\\!\\!,\,q<1$, is convergent) for all $p\in K$.
Therefore, according to Proposition 3.7, the series
$\sum_{l=0}^{\infty}r_{l}\,p^{l}$ converges uniformly and absolutely on $K$.
This proves the second statement of this proposition. The first statement
follows from the second, since any given point $p^{{}^{\prime}}\in
B\left(0,\zeta\right)$ is contained in some ball
$B\left(0,\zeta^{{}^{\prime}}\right)=\left\\{p\\!:\mid\\!p\mid<\zeta^{{}^{\prime}}\right\\}\\!,\,\,\mid\\!p^{{}^{\prime}}\\!\\!\mid<\zeta^{{}^{\prime}}<\zeta,$
belonging compactly to
$B\left(0,\zeta\right)=B\left(0,\mid\\!p_{0}\\!\mid\right)$. ∎

The quaternionic sets $G_{4}$ can be regarded as compact, since we assume that
they are continuous and bounded; every sequence in them has a subsequence that
converges to an element (a point) contained again in them.

In complex analysis the following theorem (Weierstrass) [9, p. 106] holds
true:
If the functional series

$f\left(\xi\right)=\sum_{l=0}^{\infty}f_{l}\left(\xi\right)\\!,\,\,\,\,\xi\in\mathbb{C}$
(20)

of C-holomorphic functions $f_{l}$ in an open connected domain
$D\left(\xi\right)$ converges uniformly on any compact subset of this domain,
then the sum $f\left(\xi\right)$ of this series is holomorphic in
$D\left(\xi\right)$ and the series can be differentiated termwise:

$f^{{}^{\prime}}\left(\xi\right)=\sum_{l=0}^{\infty}f_{l}^{{}^{\prime}}\left(\xi\right).$
(21)

Such a differentiation can be performed arbitrarily many times at any point
$\xi\in D\left(\xi\right)$.

Without loss of generality, we regard a domain $D\left(\xi\right)$ as a set
$G_{2}$ (Theorem 1.2) that is an open connected disk
$D\left(0,R\right)=\left\\{\xi:\mid\xi\mid<R\right\\}$, where $R>0$ is a
radius of convergence of the series (20). Correspondingly, we also regard an
open connected quaternionic ball $B\left(0,R\right)=\left\\{p\\!:\mid
p\mid<R\right\\}$ as a quaternionic generalization $G_{4}$ of a complex disk
$D\left(0,R\right)$ such that $D\left(0,R\right)$ follows from
$B\left(0,R\right)$ upon the transition from $p$ to $\xi$.

We see that the requirements for the termwise differentiation in complex
analysis are the following: (A) the functions $f_{l}\left(\xi\right)$ are
C-holomorphic in $D\left(\xi\right)$; (B) the series (20) converges uniformly
on any compact subset of $D\left(\xi\right)$. At that, we can compare radiuses
of convergence of the series before and after differentiation. If analogous
requirements will be fulfilled in the case of quaternionic functions and
series, then we can apply Theorem 1.3 and Corollary 1.4 to prove the
possibility of a termwise differentiability in the quaternionic case. Adapting
all this to the quaternionic case we formulate the following

######  Proposition 3.9.

(a termwise differentiation) If the quaternionic power series

$f\\!\left(p\right)=\sum_{l=0}^{\infty}r_{l}\,p^{l},\,\,\,\,p\in\mathbb{H},\,\,\,l=0,1,2,\dots$
(22)

of the H-holomorphic functions $r_{l}\,p^{l}$ in its open connected
convergence ball $B\left(0,R\right)=\left\\{p\\!:\mid\\!p\\!\mid<R\right\\}$
converges uniformly on any compact subset $T\in B\left(0,R\right)\\!,$ then
the sum $f\\!\left(p\right)$ of this series is H-holomorphic in
$B\left(0,R\right)$ and the series can be termwise differentiated at each
point $p\in B\left(0,R\right)$:

$f^{{}^{\prime}}\\!\left(p\right)=\sum_{l=0}^{\infty}\left(r_{l}\,p^{l}\right)^{{}^{\prime}}=\sum_{l=0}^{\infty}l\,r_{l}\,p^{l-1}=r_{1}+2r_{2}p+3r_{3}p^{2}+\cdots+lr_{l}p^{l-1}+\cdots.$
(23)

Such a differentiation can be performed arbitrarily many times at any point
$p\in B\left(0,R\right)$.

###### Proof.

It is evident, according to Theorems 1.2 , 1.3 and Corollary 1.4 , that the
function $f_{l}\left(p\right)=r_{l}\,p^{l}$ and its first derivative
$\left(r_{l}p^{l}\right)^{{}^{\prime}}=lr_{l}\,p^{l-1}$ (also derivatives of
all orders) are H-holomorphic in $B\left(0,R\right)$. Hence the requirement
(A) of complex analysis is fulfilled in the quaternionic case. The
H-holomorphy of $f\\!\left(p\right)=\sum_{l=0}^{\infty}r_{l}\,p^{l}$ in a
convergence ball $B\left(0,R\right)$ follows from Theorem 1.2 applied to the
C-holomorphic function
$\psi_{C}\\!\left(\xi\right)=\sum_{l=0}^{\infty}r_{l}\,\xi^{l}$ in its
convergence disk $D\left(0,R\right)$. We now prove that
$f\\!\left(p\right)=\sum_{l=0}^{\infty}r_{l}p^{l}$ converges uniformly on any
compact subset $T\subset B\left(0,R\right)$. Take any $p\in B\left(0,R\right)$
and consider any positive $\zeta$ such that
$\mid\\!p\\!\mid<\zeta=\mid\\!p_{0}\\!\mid<R$. By virtue of the convergence of
the series

$\sum_{l=0}^{\infty}r_{l}\,\zeta^{l}=r_{0}+r_{1}\zeta+r_{2}\zeta^{2}+\cdots+r_{l}\zeta^{l}+\cdots,\,\,\,\,\,r_{l},\zeta\in\mathbb{R},$

and in accordance with Proposition 3.5 we have $r_{l}\,\zeta^{l}\rightarrow 0$
as $l\rightarrow{\infty}.$ Then there exists a big enough constant $M>0$ such
that the general term of this series is bounded above:

$r_{l}\zeta^{l}\leq M,\,\,\,l=0,1,2,\dots.$

We have $r_{l}\\!\\!\mid\\!\\!p_{0}\\!\\!\mid^{l}=r_{l}\zeta^{l}\leq M$ and
series (22) converges absolutely and uniformly on any set $T\subset
B\left(0,\zeta\right)$ in accordance with Proposition 3.8. Hence the
requirement (B) of the complex analysis is also fulfilled in the quaternionic
case.

Now we compare radiuses of convergence of the series before and after
differentiation. For an absolute value of the $l^{\,{}^{\prime}}\\!th$ term of
the series (23) we have the following estimate:

$\mid\\!\left(r_{l}\,p^{l}\right)^{{}^{\prime}}\\!\\!\mid=lr_{l}\mid\\!p^{l-1}\\!\mid=lr_{l}\zeta^{l}\cdot\mid\\!\frac{p}{\zeta}\\!\mid^{l-1}\cdot\frac{1}{\zeta}\leq\frac{M}{\zeta}\cdot
l\cdot\mid\\!\frac{p}{\zeta}\\!\mid^{l-1},\,\,\,\,\,\,l=0,1,2,\dots,$

where the expression for the first derivative:
$\left(r_{l}\,p^{l}\right)^{{}^{\prime}}\\!\\!=lr_{l}p^{l-1}$ is used. The
series

$\sum_{l=0}^{\infty}\mid\\!\left(r_{l}\,p^{l}\right)^{{}^{\prime}}\\!\\!\mid\leq\frac{M}{\zeta}\sum_{l=0}^{\infty}l\mid\\!\frac{p}{\zeta}\\!\mid^{l-1}=\frac{M}{\zeta}\left\\{1+2\mid\\!\frac{p}{\zeta}\\!\mid+3\mid\\!\frac{p}{\zeta}\\!\mid^{2}+\cdots+l\mid\\!\frac{p}{\zeta}\\!\mid^{l-1}+\cdots\right\\}$

is absolutely convergent in $B\left(0,\zeta\right)$. Given
$\mid\frac{p}{\zeta}\mid<1$, one can verify this fact by using Proposition 3.6
. Then, by Proposition 3.7, the series
$\sum_{l=0}^{\infty}\left(r_{l}\,p^{l}\right)^{{}^{\prime}}$ converges
absolutely and uniformly on $B\left(0,\zeta\right)$. Since the series
$\sum_{l=0}^{\infty}\left(r_{l}\,p^{l}\right)^{{}^{\prime}}$ converges
absolutely in $B\left(0,\zeta\right)$ and the choice of $p_{0}$ is arbitrary,
it is clear that the convergence radius $R^{{}^{\prime}}$ of the series (23)
is not less than the convergence radius $R$ of the series (22). This
conclusion is sufficient for our objective and we don’t need to prove that
$R^{{}^{\prime}}$ is exactly equal to $R$. By repeating this reasoning , we
prove that the series (22) can be differentiated termwise arbitrarily many
times in its ball of convergence. ∎

######  Proposition 3.10.

(a quaternionic Maclaurin series). Let a H-holomorphic function
$\psi_{H}\\!\left(p\right)$ be represented by a sum of a convergent
quaternionic power series:

$\psi_{H}\\!\left(p\right)=\sum_{k=0}^{\infty}r_{k}\,p^{k}=r_{0}+r_{1}p+r_{2}p^{2}+\cdots+r_{k}p^{k}+\cdots,\,\,\,\,r_{k}\in\mathbb{R}$

in a ball $B\left(0,R\right)=\left\\{p\\!:\mid\\!p\\!\mid<R\right\\}$.Then the
coefficients $r_{k}$ are determined uniquely as

$r_{k}=\frac{\psi_{H}^{\left(k\right)}\\!\\!\left(0\right)}{k!},\,\,\,\,k=0,1,2,\dots,$
(24)

where $\psi_{H}^{\left(k\right)}\\!\\!\left(0\right)\\!,k=1,2,3,\dots$ is the
$k^{{}^{\prime}}th$ full derivative of $\psi_{H}\\!\left(p\right)$ at $p=0$
and $\psi_{H}^{\left(0\right)}\\!\\!\left(0\right)$ is
$\psi_{H}\\!\left(0\right)$.

###### Proof.

Since $\psi_{H}\\!\left(p\right)$ is H-holomorphic in a ball
$B\left(0,R\right)$, there exist, according to Theorem 1.2, all derivatives
$\psi_{H}^{\left(k\right)}\\!\\!\left(0\right),\,k=1,2,\dots,$ in
$B\left(0,R\right)$. Putting $p=0$ in
$\psi_{H}\\!\left(p\right)=\sum_{k=0}^{\infty}r_{k}\,p^{k}$ we find
$r_{0}=\psi_{H}\\!\left(0\right)$. Differentiating
$\psi_{H}\\!\left(p\right)=\sum_{k=0}^{\infty}r_{k}\,p^{k}$ termwise and using
the formula $\left(r_{k}\,p^{k}\right)^{{}^{\prime}}=r_{k}kp^{k-1}$, we obtain

$\psi_{H}^{{}^{\prime}}\\!\left(p\right)=r_{1}+2r_{2}p+3r_{3}p^{2}+\cdots+kr_{k}p^{k-1}+\cdots.$

Putting $p=0$ into this expression, we obtain
$r_{1}=\psi_{H}^{{}^{\prime}}\\!\left(0\right)$. Further, differentiating the
expression for $\psi_{H}^{{}^{\prime}}\\!\left(p\right)$, we get the second
derivative as follows:

$\psi_{H}^{\left(2\right)}\\!\left(p\right)=2r_{2}+3\cdot 2r_{3}p+4\cdot
3r_{4}p^{2}+5\cdot
4r_{5}p^{3}+\cdots+k\cdot\left(k-1\right)r_{k}p^{k-2}+\cdots.$

Putting $p=0$ into this expression, we obtain
$r_{2}=\frac{\psi_{H}^{\left(2\right)}\\!\left(0\right)}{2}.$ Differentiating
$\psi_{H}\\!\left(p\right)=\sum_{k=0}^{\infty}r_{k}\,p^{k}\,\,k$ times, we get

$\psi_{H}^{\left(k\right)}\\!\left(p\right)=k!r_{k}+positive\,powers\,of\,p\,multiplied\,by\,integers.$

Then once again putting $p=0$, we obtain
$r_{k}=\frac{\psi_{H}^{\left(k\right)}\\!\\!\left(0\right)}{k\,!}.$ ∎

Note that representations of H-holomorphic functions by Maclaurin series are
unique, since the coefficients of series are determined uniquely by formula
(24) when using the full quaternionic derivatives (6) of the functions in
question. Therefore, in a problem of representing a H-holomorphic function by
a quaternionic power series the answer does not depend on the methods adopted
for this purpose.

We can introduce the notion of a H-analytic function by the following

######  Definition 3.11.

(a H-analytic function) A quaternionic function is said to be H-analytic in an
open connected ball $B\left(0,R\right)$ if it can be expanded as a convergent
Maclaurin (Taylor) series in $B\left(0,R\right)$, where $R\geq 0$ is a radius
of convergence.

Whether we speak of a Maclaurin or a Taylor series, doesn’t matter here, since
we can go from one series to another by replacing variables. Now we can
formulate the following

######  Proposition 3.12.

Every H-holomorphic function is H-analytic.

###### Proof.

According to Proposition 3.10, every H-holomorphic function can be uniquely
expanded as a convergent Maclaurin (Taylor) series in some open connected ball
$B\left(0,R\right)$, where $R\geq 0$ is a radius of series convergence. ∎

######  Proposition 3.13.

(a similarity between C- and H-representations of power series). The power
series expansions of every holomorphic function in C- and H-representations
are similar and converge with identical convergence radiuses $R$. In a
C-representation we regard an $R$ as a radius of a convergence disk and in a
H-representation as a radius of a corresponding convergence ball.

###### Proof.

According to Theorem 1.2 , every H-holomorphic function (also its series in
H-representation) defined in an open connected domain
$G_{4}\subseteq\mathbb{H}$ is constructed from a C-holomorphic function of the
same kind (also its series in C-representation) defined in an open connected
domain $G_{2}\subseteq\mathbb{C}$ by replacing a complex variable by a
quaternionic one, and $G_{4}$ is such that $G_{2}$ exactly follows from
$G_{4}$ upon the transition from $p$ to $\xi$. Given this relationship, if we
regard, as mentioned above, a domain $G_{2}$ as a complex disk
$D\left(0,R\right)=\left\\{\xi:\mid\xi\mid<R\right\\}$, then a domain $G_{4}$
can only be a quaternionic ball
$B\left(0,R\right)=\left\\{p\\!:\mid\\!p\mid<R\right\\}$ with the same $R$,
i.e. the convergence radiuses of the series of a holomorphic function in C-
and H-representstions are equal. The similarity of series (the same
coefficients of the terms of the same degrees) in C- and H-representations
also follows from Theorem 1.2. ∎

The quaternionic generalization of Laplace’s equations within the theory of
H-holomorphic functions is presented in [11, p. 5]. At that, H-harmonic
components of H-holomorphic functions are denoted by
$\psi_{1}\left(x,y,z,u\right),\,\psi_{2}\left(x,y,z,u\right),\,\psi_{3}\left(x,y,z,u\right)$
and $\psi_{4}\left(x,y,z,u\right).$

Note that if coefficients of an initial complex series contain explicitly the
imaginary unit $i$, we must use replacing [5, p. 6] the imaginary complex
unit $i$ with the quaternionic imaginary unit $r$ in order to obtain a
quaternionic analogue of this complex series.

##  4 Examples

###  4.1 Examples of the elementary functions $e^{p},\,\sin p,\cos p$

In complex analysis we have, for example, the following power series
expansions [8]:

$\displaystyle
e^{\xi}=\sum_{l=0}^{\infty}\frac{\xi^{l}}{l\,!}=1+\xi+\frac{\xi^{2}}{2\,!}+\frac{\xi^{3}}{3\,!}+\cdots,\,\,\,\xi\in\mathbb{C},$
$\displaystyle\sin\xi=\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\frac{\xi^{2l+1}}{\left(2l+1\right)!}=\xi-\frac{\xi^{3}}{3!}+\frac{\xi^{5}}{5!}-\frac{\xi^{7}}{7!}+\cdots,$
$\displaystyle\cos\xi=\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\frac{\xi^{2l}}{\left(2l\right)!}=1-\frac{\xi^{2}}{2!}+\frac{\xi^{4}}{4!}-\frac{\xi^{6}}{6!}+\cdots.$

Replacing $\xi$ by $p$ in these expressions, we obtain in accordance with
Theorem 1.2 the following power series expansions about a point $p=0$ for the
H-holomorphic functions $e^{p},\,\sin p,$ and $\cos p:$

$e^{p}=\sum_{l=0}^{\infty}\frac{p^{l}}{l!}=1+p+\frac{p^{2}}{2\,!}+\frac{p^{3}}{3\,!}+\cdots,\,\,\,p\in\mathbb{H},$
(25) $\sin
p=\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\frac{p^{2l+1}}{\left(2l+1\right)!}=p-\frac{p^{3}}{3!}+\frac{p^{5}}{5!}-\frac{p^{7}}{7!}+\cdots,$
(26) $\cos
p=\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\frac{p^{2l}}{\left(2l\right)!}=1-\frac{p^{2}}{2!}+\frac{p^{4}}{4!}-\frac{p^{6}}{6!}+\cdots.$
(27)

The verification of the H-holomorphy of the functions $\sin p$ and $\cos p$ on
the left-hand of these expressions can be made analogously to those of
H-holomorphy of the function $e^{p}\\!,$ which is given in [5, p. 6]. We
skip the details.

In accordanse with Proposition 3.1 we can speak of a H-representation of the
functions $exp,sin$ and $cos.$

Applying d’Alembert’s ratio test (Proposition 3.6) to the series
$e^{p}=\sum_{l=0}^{\infty}r_{l}\,p^{l},$ where $r_{l}=\frac{1}{l!},$ we have

$L=\lim_{l\rightarrow\infty}\frac{\mid\\!r_{\left(l+1\right)}\,p^{{l+1}}\\!\mid}{\mid\\!r_{l}\,p^{l}\\!\mid}=\lim_{l\rightarrow\infty}\frac{\mid
p^{{l+1}}\\!\mid
l\,!}{\left(l+1\right)!\mid\\!p^{l}\\!\mid}=\mid\\!p\\!\mid\\!\\!\lim_{l\rightarrow\infty}\frac{1}{l+1}=0<1.$

Then, in accordance with this test, the radius of convergence of the series
$e^{p}=\sum_{l=0}^{\infty}\frac{p^{l}}{l\,!}$ is $R=\frac{1}{L}=\infty.$

Analogously, for the series $\sin
p=\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\frac{p^{{2l+1}}}{\left(2l+1\right)!}$
we find

$\displaystyle L$
$\displaystyle=\lim_{l\rightarrow\infty}\frac{\mid\\!r_{\left(l+1\right)}\,p^{{l+1}}\\!\mid}{\mid\\!r_{l}\,p^{l}\\!\mid}=\lim_{l\rightarrow\infty}\frac{\left(-1\right)^{l+1}\mid
p^{2\left(l+1\right)+1}\\!\mid\left(2l+1\right)!}{\left[2\left(l+1\right)+1\right]!\,\left(-1\right)^{l}\mid\\!p^{{2l+1}}\\!\mid}$
$\displaystyle=-\lim_{l\rightarrow\infty}\frac{\mid
p^{2}\\!\mid}{\left(2l+2\right)\left(2l+3\right)}=0<1,$

whence the radius of convergence of the power series expansion of $\sin p$ is
also $R=\frac{1}{L}=\infty.$ The calculation for $\cos p$ gives $R=\infty$ as
well. Thus, the convergence set of series expansions for the functions
$e^{p}\\!,\,\sin p,\cos p$ is the set of all quaternion numbers. We see that
these results correspond Proposition 3.13 and we further can immediately get a
convergence radiuses in similar cases without making any calculations.

The verification of the uniform convergence of these series is analogous to
those used in the proof of Proposition 3.9, since these series can be
majorized for any $p\in\mathbb{H}$ by the convergent series with the terms
$r_{l}\,\zeta^{l},$ where $\zeta$ is a positive number such that
$\mid\\!p\\!\mid<\zeta<\infty.$ According to Proposition 3.9 , we can
differentiate termwise these series at all points $p\in\mathbb{H}.$ At that,
we get the following expressions for derivatives:

$\displaystyle\left(e^{p}\right)^{{}^{\prime}}=\left(\sum_{l=0}^{\infty}\frac{p^{l}}{l!}\right)^{{}^{\prime}}=\sum_{l=0}^{\infty}\left(\frac{p^{l}}{l!}\right)^{{}^{\prime}}=\sum_{k=1}^{\infty}\frac{p^{k-1}}{\left(k-1\right)!}=\sum_{l=0}^{\infty}\frac{p^{l}}{l!}=e^{p},$
$\displaystyle\left(\sin
p\right)^{{}^{\prime}}=\left(\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\frac{p^{2l+1}}{\left(2l+1\right)!}\right)^{{}^{\prime}}=\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\left(\frac{p^{2l+1}}{\left(2l+1\right)!}\right)^{{}^{\prime}}$
$\displaystyle=\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\frac{p^{2l}}{\left(2l\right)!}=\cos
p,$ $\displaystyle\left(\cos
p\right)^{{}^{\prime}}=\left(\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\frac{p^{2l}}{\left(2l\right)!}\right)^{{}^{\prime}}=\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\left(\frac{p^{2l}}{\left(2l\right)!}\right)^{{}^{\prime}}$
$\displaystyle=\sum_{k=1}^{\infty}{\left(-1\right)}^{k}\frac{p^{2k-1}}{\left(2k-1\right)!}=-\sum_{l=0}^{\infty}{\left(-1\right)}^{l}\frac{p^{{2l+1}}}{\left(2l+1\right)!}=-\sin
p,$

which are analogous to those in complex analysis.

Now we want to show that formulae (25), (26) and (27) can be obtained by means
of Maclaurin series expansions (Proposition 3.10) of the H-holomorphic
functions $e^{p},\,\sin p,$ and $\cos p.$ In other words, we want to show that
the coefficients of these series, which are preserved during the transition
from the complex to the quaternion case by the replacement of a complex
variable by a quaternionic one, are coefficients (24) of Maclaurin series
expansions of these functions.

We can 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

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

is a purely imaginary unit quaternion, so its square is $-1.$ 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 $\xi=x+yi.$ Then we generalize
the complex Euler formula [8] as $e^{{rV}}=\cos V+r\sin V,$ and then the known
complex formulae $\cos\xi=\frac{e^{{i\xi}}+e^{{-i\xi}}}{2}$ and
$\sin\xi=\frac{e^{{i\xi}}-e^{{-i\xi}}}{{2i}}$ as :

$\cos p=\frac{e^{{rp}}+e^{{-rp}}}{2},$ (28) $\sin
p=\frac{e^{{rp}}-e^{{-rp}}}{{2r}},$ (29)

where the complex imaginary unit $i$ is replaced [5, p. 6] by the
quaternionic $r$.

Using (28), (29) and relations
$x=\frac{a+\overline{a}}{2},\,\,y=\frac{a-\overline{a}}{{2i}},\,\,z=\frac{b+\overline{b}}{2},\,\,u=\frac{b-\overline{b}}{{2i}},$
following from (2), we obtain after some cumbersome calculation the following
expressions for the functions $\cos p$ and $\sin p$ in the Cayley–Dickson
doubling form:

$\cos p=\frac{e^{{rp}}+e^{{-rp}}}{2}=\Phi_{{1\left(\cos
p\right)}}+\Phi_{{2\left(\cos p\right)}}\cdot j,$

where

$\Phi_{{1\left(\cos
p\right)}}=\frac{\left(e^{{-V}}+e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V}},$
(30) $\Phi_{{2\left(\cos
p\right)}}=\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{2V}}b$
(31)

and

$\sin p=\frac{e^{{rp}}-e^{{-rp}}}{{2r}}=\Phi_{{1\left(\sin
p\right)}}+\Phi_{{2\left(\sin p\right)}}\cdot j,$

where (see also [4, p. 19])

$\Phi_{{1\left(\sin
p\right)}}=\frac{\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2}-\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{4V}},$
(32) $\Phi_{{2\left(\sin
p\right)}}=-\frac{\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{2V}}b.$
(33)

According to [5, p. 5], we have the function
$e^{p}=\Phi_{{1\left(e^{p}\right)}}+\Phi_{{2\left(e^{p}\right)}}\cdot j$,
where

$\Phi_{{1\left(e^{p}\right)}}=2\beta\cos
V+\frac{\beta\left(a-\overline{a}\right)\sin V}{V},$ (34)
$\Phi_{{2\left(e^{p}\right)}}=\frac{2\beta\sin V}{V}b,$ (35)

and $\beta=\frac{e^{\frac{a+\overline{a}}{2}}}{2}.$

Now we calculate the values of the functions $e^{p},\,\sin p,$ and $\cos p$ as
well as the value of $V=\sqrt{y^{2}+z^{2}+u^{2}}$ at the point $p=a+b\cdot
j=0,$ i.e. at

$a=0,\,\,\,\,\,\,\overline{a}=0,\,\,\,\,\,\,b=0,\,\,\,\,\,\,\overline{b}=0\,\,\left(\text{or}\,\,x=y=z=u=0\right),\,\,\,\,\,\,V=0.$
(36)

At that, to eliminate indeterminate expressions of the form $\frac{0}{0}$ we
use the known limit relation [8, p. 233]:

$\lim_{V\rightarrow 0}\frac{\sin V}{V}=1$ (37)

as well as the following limit relation calculated by using L’Hospital’s rule
[10, p. 254]:

$\lim_{V\rightarrow 0}=\frac{e^{{-V}}-e^{V}}{V}=\lim_{V\rightarrow
0}\frac{\left(e^{{-V}}-e^{V}\right)^{{}^{\prime}}}{V^{{}^{\prime}}}=\lim_{V\rightarrow
0}\frac{{-e}^{{-V}}-e^{V}}{1}=-2.$ (38)

Substituting (36), (37), (38) into (30), (31), (32), (33), (34) and (35), we
get the following values for the functions $e^{p},\,\sin p,$ and $\cos p$ at
the point $p=0:$

$e^{0}=1,\,\,\,\,\sin 0=0,\,\,\,\,\cos 0=1,$ (39)

coinciding with analogous values of the corresponding functions in real and
complex areas. Further, using (39), we get the expansion coefficients for
series (22) when $l=0$ as follows:

$r_{0\left(e^{p}\right)}=e^{0}=1,\,\,\,\,r_{0\left(\sin p\right)}=\sin
0=0,\,\,\,\,r_{0\left(\cos p\right)}=\cos 0=1.$

They coincide with the coefficients of the terms of degree 0 in (25), (26),
and (27).

As shown in [5, p. 7], the derivatives of all orders of the quaternionic
function $e^{p}$ are equal to the function itself. Then, according to (24), we
get
$r_{k\left(e^{p}\right)}=\frac{\psi_{H}^{\left(k\right)}\\!\\!\left(0\right)}{k\,!}=\frac{\left(e^{p}\right)^{{\left(k\right)}}\left(0\right)}{{k!}}=\frac{e^{0}}{{k!}}=\frac{1}{{k!}},$
where $k=0,1,2,\dots.$ The obtained values of coefficients coincide with the
corresponding values of coefficients in (25). The Maclaurin series expansion
for the function $e^{p}$ coincides with series (25) obtained by using Theorem
1.2.

Now, we show how the derivatives of $\cos p$ and $\sin p$ are further computed
within the framework of H-holomorphic functions theory. According to (4), we
define the first full quaternionic derivative of $\cos p$ as follows:

$\left(\cos p\right)^{{}^{\prime}}=\Phi_{{1\left(\cos
p\right)}}^{{}^{\prime}}+\Phi_{{2\left(\cos p\right)}}^{{}^{\prime}}\cdot j,$
(40)

where

$\Phi_{{1\left(\cos p\right)}}^{{}^{\prime}}=\partial_{a}\Phi_{{1\left(\cos
p\right)}}+\partial_{\overline{a}}\Phi_{{1\left(\cos p\right)}},$ (41)
$\Phi_{{2\left(\cos p\right)}}^{{}^{\prime}}=\partial_{a}\Phi_{{2\left(\cos
p\right)}}+\partial_{\overline{a}}\Phi_{{2\left(\cos p\right)}}.$ (42)

Now we need to compute the partial derivatives of $\Phi_{{1\left(\cos
p\right)}}$ and $\Phi_{{2\left(\cos p\right)}}$ with respect to each of
$a,\overline{a}.$ After some cumbersome calculations, we have for
$\partial_{a}\Phi_{{1\left(\cos p\right)}}$ and
$\partial_{\overline{a}}\Phi_{{1\left(\cos p\right)}}$ the following
expressions:

$\begin{split}\partial_{a}\Phi_{{1\left(\cos
p\right)}}=\partial_{a}&\left[\frac{\left(e^{{-V}}+e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V}}\right]\\\
=\frac{1}{2}&\left[\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{4V}}-\frac{\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2}\right]\\\
+\frac{1}{2}&\left[\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2V}+\frac{\left(a-\overline{a}\right)^{2}\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{8V^{2}}\right.\\\
&\left.+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{4V}+\frac{\left(a-\overline{a}\right)^{2}\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{8V^{3}}\right]\\!,\end{split}$
$\begin{split}\partial_{\overline{a}}\Phi_{{1\left(\cos
p\right)}}=\partial_{\overline{a}}&\left[\frac{\left(e^{{-V}}+e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V}}\right]\\\
=\frac{1}{2}&\left[-\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{4V}}-\frac{\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2}\right]\\\
+\frac{1}{2}&\left[-\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2V}-\frac{\left(a-\overline{a}\right)^{2}\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{8V^{2}}\right.\\\
&\left.+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{4V}-\frac{\left(a-\overline{a}\right)^{2}\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{8V^{3}}\right]\\!.\\\
\end{split}$

Substituting these expressions into (41) and reducing like terms, we get

$\Phi_{{1\left(\cos
p\right)}}^{{}^{\prime}}=-\frac{\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{4V}}=-\Phi_{{1\left(\sin
p\right)}}.$ (43)

Quite analogously we obtain for the partial derivatives
$\partial_{a}\Phi_{{2\left(\cos p\right)}}$ and
$\partial_{\overline{a}}\Phi_{{2\left(\cos p\right)}}$ the following
expressions:

$\begin{split}\partial_{a}\Phi_{{2\left(\cos
p\right)}}&=\frac{b}{2}\,\partial_{a}\left[\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{V}\right]=\frac{b}{2}\left[\frac{\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2V}\right.\\\
&+\left.\frac{\left(a-\overline{a}\right)\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V^{2}}}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V^{3}}}\right],\\\
\end{split}$ $\begin{split}\partial_{\overline{a}}\Phi_{{2\left(\cos
p\right)}}&=\frac{b}{2}\,\partial_{\overline{a}}\left[\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{V}\right]=\frac{b}{2}\left[\frac{\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2V}\right.\\\
&-\left.\frac{\left(a-\overline{a}\right)\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V^{2}}}-\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V^{3}}}\right].\\\
\end{split}$

Substituting these expressions into (42) we get the second constituent of the
full quaternionic derivative of $\cos p$:

$\Phi_{{2\left(\cos p\right)}}^{{}^{\prime}}=\partial_{a}\Phi_{{2\left(\cos
p\right)}}+\partial_{\overline{a}}\Phi_{{2\left(\cos
p\right)}}=\frac{\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2V}b=-\Phi_{{2\left(\sin
p\right)}}.$ (44)

Next, substituting (43) and (44) into (40), we get the following expression:

$\left(\cos p\right)^{{}^{\prime}}=\Phi_{{1\left(\cos
p\right)}}^{{}^{\prime}}+\Phi_{{2\left(\cos p\right)}}^{{}^{\prime}}\cdot
j=-\Phi_{{1\left(\sin p\right)}}-\Phi_{{2\left(\sin p\right)}}\cdot j=-\sin
p.$ (45)

Thus, we also have shown that the derivatives of the quaternionic and complex
cosine functions have indeed identical expressions in accordance with
Corollary 1.4.

In an analogous fashion, we obtain for the quaternionic derivative of the
function $\sin p$ the following expression:

$\left(\sin p\right)^{{}^{\prime}}=\cos p,$ (46)

which coincides with the corresponding expression from the complex analysis.
We skip the details here.

Formulae (45) and (46) allow us further to calculate, using (24) , the
coefficients
$r_{k}=\frac{\psi_{H}^{\left(k\right)}\\!\\!\left(0\right)}{k\,!},\,\,\,\,k=0,1,2,\dots,$
of Maclaurin series expansions for the functions $\sin p$ and $\cos p$ to show
that they coincide with corresponding coeffitients in (26) and (27).

To illustrate this we consider the function $\sin p.$ For $\sin p$ we get the
coefficient of $p^{1}$ as follows:
$r_{1}=\frac{\psi_{H}^{\left(1\right)}\\!\\!\left(0\right)}{1\,!}=\left(\sin
p\right)_{p=0}^{{\left(1\right)}}=\cos 0=1.$ It coincides with the coefficient
of $p^{1}$ in (26) that equals 1. The calculation of the coefficient of
$p^{2}$ gives $r_{2\left(\sin
p\right)}=\frac{\psi_{H}^{\left(2\right)}\\!\\!\left(0\right)}{2!}=\frac{\left(\sin
p\right)_{p=0}^{\left(2\right)}}{2}=\frac{\left(\cos
p\right)_{p=0}^{\left(1\right)}}{2}=-\frac{\sin 0}{2}=0.$ It coincides with
the coefficient of $p^{2}$ in (26) that is equal to 0. The coefficient of
$p^{3}$ is calculated as follows: $r_{3\left(\sin
p\right)}=\frac{\psi_{H}^{\left(3\right)}\\!\\!\left(0\right)}{3!}=\frac{\left(\sin
p\right)_{p=0}^{\left(3\right)}}{3!}=-\frac{\cos 0}{3!}=-\frac{1}{3!}.$ It
coincides with the coefficient of $p^{3}$ in (26) that is equal to
$-\frac{1}{{3!}}.$ In this way, we can go on.

We have shown that the application of Theorems 1.2, 1.3, and Corollary 1.4 to
infinite series gives true results.

###  4.2 Example of a product of H-holomorphic functions

Consider a more complicated example: a quaternionic product of two
H-holomorphic functions $\sin p$ and $\cos p$. Since a qiaternionic product of
H-holomorphic functions is also a H-holomorphic function (see [4, p. 18]),
we can, using (6), calculate the full quaternionic derivative of each order
and then, using (24), obtain each coefficient $r_{k}$ of a Maclaurin series
expansion for the quaternionic product in question. However, such a
calculation is cumbersome and we can use in accordance with Corollary 1.4 the
following product differentiation rule (see also [4, p. 20]):
${{\left(f\cdot g\right)}}^{{}^{\prime}}=f^{{}^{\prime}}\cdot g+f\cdot
g^{{}^{\prime}},$ where $f$ and $g$ are arbitrary H-holomorphic functions.

To establish a general expression for terms of this series we calculate a
sufficient number of the terms and derivatives. Given (45) and (46), we obtain
for the derivatives of the function $\psi_{H}\\!\left(p\right)=\sin p\cdot\cos
p$ the following expressions:

$\displaystyle\psi_{H}^{{\left(1\right)}}\\!\left(p\right)=\cos^{2}p-\sin^{2}p,\,\,\,\,\psi_{H}^{{\left(2\right)}}\\!\left(p\right)=-4\sin
p\,\cos
p,\,\,\,\,\psi_{H}^{{\left(3\right)}}\\!\left(p\right)=-4\left(\cos^{2}p-\sin^{2}p\right),$
$\displaystyle\psi_{H}^{{\left(4\right)}}\\!\left(p\right)=16\sin p\,\cos
p,\,\,\,\,\psi_{H}^{{\left(5\right)}}\\!\left(p\right)=16\left(\cos^{2}p-\sin^{2}p\right),\,\,\,\,\psi_{H}^{{\left(6\right)}}\\!\left(p\right)=-64\sin
p\,\cos p,$
$\displaystyle\psi_{H}^{{\left(7\right)}}\\!\left(p\right)=-64\left(\cos^{2}p-\sin^{2}p\right)\\!,\,\psi_{H}^{{\left(8\right)}}\\!\left(p\right)=256\sin
p\,\cos
p,\,\psi_{H}^{{\left(9\right)}}\\!\left(p\right)=256\left(\cos^{2}p-\sin^{2}p\right)\\!,$
$\displaystyle\psi_{H}^{{\left(10\right)}}\\!\left(p\right)=-1024\sin p\,\cos
p,\,\,\,\psi_{H}^{{\left(11\right)}}\\!\left(p\right)=-1024\left(\cos^{2}p-\sin^{2}p\right)\\!,\,$
$\displaystyle\psi_{H}^{{\left(12\right)}}\\!\left(p\right)=4096\sin p\,\cos
p,\,\,\,\psi_{H}^{{\left(13\right)}}\\!\left(p\right)=4096\left(\cos^{2}p-\sin^{2}p\right),\,\,\,\,$
$\displaystyle\psi_{H}^{{\left(14\right)}}\\!\left(p\right)=-16384\sin p\,\cos
p,\,\,\,\,\psi_{H}^{{\left(15\right)}}\\!\left(p\right)=-16384\left(\cos^{2}p-\sin^{2}p\right),$
$\displaystyle\psi_{H}^{{\left(16\right)}}\\!\left(p\right)=65536\sin p\,\cos
p,\,\,\,\,\psi_{H}^{{\left(17\right)}}\\!\left(p\right)=65536\left(\cos^{2}p-\sin^{2}p\right).$

Substituting $p=0$ into these expressions and using (24), we get the
coeffitients of the Maclaurin series expansion of the function
$\psi_{H}\\!\left(p\right)=\sin p\cdot\cos p$ as follows:

$\displaystyle
r_{0}=\frac{\psi_{H}\\!\left(0\right)}{{0!}}=\frac{0}{{1}}=0,\,\,r_{1}=\frac{\psi_{H}^{{\left(1\right)}}\left(0\right)}{{1!}}=\frac{\left(\cos^{2}0-\sin^{2}0\right)}{{1!}}=\frac{1}{{1!}}=1,\,\,r_{2}=\frac{\psi_{H}^{{\left(2\right)}}\left(0\right)}{{2!}}=0,$
$\displaystyle
r_{3}=\frac{\psi_{H}^{{\left(3\right)}}\left(0\right)}{{3!}}=-\frac{4}{{3!}},\,\,r_{4}=\frac{\psi_{H}^{{\left(4\right)}}\left(0\right)}{{4!}}=0,\,\,r_{5}=\frac{\psi_{H}^{{\left(5\right)}}\left(0\right)}{{5!}}=\frac{16}{{5!}},\,\,r_{6}=\frac{\psi_{H}^{{\left(6\right)}}\left(0\right)}{{6!}}=0,$
$\displaystyle
r_{7}=\frac{\psi_{H}^{{\left(7\right)}}\left(0\right)}{{7!}}=-\frac{64}{{7!}},\,\,r_{8}=\frac{\psi_{H}^{{\left(8\right)}}\left(0\right)}{{8!}}=0,\,\,r_{9}=\frac{\psi_{H}^{{\left(9\right)}}\left(0\right)}{{9!}}=\frac{256}{{9!}},\,\,r_{10}=\frac{\psi_{H}^{{\left(10\right)}}\left(0\right)}{{10!}}=0,$
$\displaystyle
r_{11}=\frac{\psi_{H}^{{\left(11\right)}}\left(0\right)}{{11!}}=-\frac{1024}{{11!}},\,\,r_{12}=\frac{\psi_{H}^{{\left(12\right)}}\left(0\right)}{{12!}}=0,\,\,r_{13}=\frac{\psi_{H}^{{\left(13\right)}}\left(0\right)}{{13!}}=\frac{4096}{{13!}},$
$\displaystyle
r_{14}=\frac{\psi_{H}^{{\left(14\right)}}\left(0\right)}{{14!}}=0,\,\,r_{15}=\frac{\psi_{H}^{{\left(15\right)}}\left(0\right)}{{15!}}=-\frac{16384}{{15!}},\,\,r_{16}=\frac{\psi_{H}^{{\left(16\right)}}\left(0\right)}{{16!}}=0,$
$\displaystyle
r_{17}=\frac{\psi_{H}^{{\left(17\right)}}\left(0\right)}{{17!}}=\frac{65536}{{17!}}.$

Thus, we get the Maclaurin series expansion for the function
$\psi_{H}\\!\left(p\right)=\sin p\cdot\cos p$ as follows:

$\displaystyle\sin p\cdot\cos p$
$\displaystyle=p-\frac{4}{{3!}}p^{3}+\frac{16}{{5!}}p^{5}-\frac{64}{{7!}}p^{7}+\frac{256}{{9!}}p^{9}$
$\displaystyle-\frac{1024}{{11!}}p^{11}+\frac{4096}{{13!}}p^{13}-\frac{16384}{{15!}}p^{15}+\frac{65536}{{17!}}p^{17}-\cdots$

In a general form this series is the following:

$\sin p\cdot\cos=\sum_{l=0}^{\infty}\alpha_{l},$ (47)

where

$\alpha_{l}={{\left(-1\right)}}^{l}\frac{4^{l}p^{{2l+1}}}{{\left(2l+1\right)!}}.$
(48)

To verify a convergence of series (47) we use d’Alembert’s Ratio Test (see
Proposition 3.6). Replacing $l$ by $l+1$ in (48) we get the following
expression:

$\alpha_{{l+1}}={{\left(-1\right)}}^{{l+1}}\frac{4^{{l+1}}p^{{2l+3}}}{{\left(2l+3\right)!}}.$
(49)

Now, using (48) and (49) and reducing like terms, we get the quotient of
$\alpha_{{l+1}}$ by $\alpha_{{l}}$:

$\frac{\alpha_{{l+1}}}{\alpha_{l}}=\frac{{{\left(-1\right)}}^{{l+1}}\frac{4^{{l+1}}p^{{2l+3}}}{{\left(2l+3\right)!}}}{{{\left(-1\right)}}^{l}\frac{4^{l}p^{{2l+1}}}{{\left(2l+1\right)!}}}=-\frac{4p^{2}}{\left(2l+2\right)\left(2l+3\right)}.$
(50)

Further, taking limit (18), we have as follows:

$\lim_{l\rightarrow\infty}=\frac{\mid\\!\alpha_{{l+1}}\\!\mid}{\mid\\!\alpha_{l}\\!\mid}=L=\lim_{l\rightarrow\infty}\frac{4\mid\\!p^{2}\\!\mid}{\left(2l+2\right)\left(2l+3\right)}=0,$
(51)

i.e. the series (47) is absolutely convergent at all $p\in\mathbb{H}.$

To check whether series (47) is uniformly convergent at all $p\in\mathbb{H}$
we can use the value $M_{l}=\frac{5^{l}p^{{2l+1}}}{{\left(2l+1\right)!}}$ such
that
$\mid\\!\\!\alpha_{l}\\!\\!\mid=\frac{4^{l}p^{{2l+1}}}{{\left(2l+1\right)!}}\leq
M_{l}$ for $l=0,1,2,\dots$ and at all $p\in\mathbb{H}.$ At that, the series
$\sum_{l=0}^{\infty}M_{l}$ is absolutely convergent at all $p\in\mathbb{H}$ in
accordance with Proposition 3.6, since $\lim_{l\rightarrow\infty}\frac{\mid
M_{{l+1}}\mid}{\mid M_{l}\mid}=\lim_{l\rightarrow\infty}\frac{5\mid
p\mid^{2}}{{\left(2l+2\right)\left(2l+3\right)}}=0.$

Thus, according to Proposition 3.7, series (47) converges absolutely and
uniformly at all $p\in\mathbb{H}$.

Bearing in mind the paramount importance of the isomorphism between the C- and
H-representations for the theory in question, we would also like to
demonstrate by a direct calculation that the quaternionic multiplication of
the H-holomorphic functions $\sin p$ and $\cos p$ behaves as commutative. In
the case of multiplication of two arbitrary quaternionic functions
$f\\!\left(p\right)=f_{1}+f_{2}\cdot j$ and
$g\\!\left(p\right)=g_{1}+g_{2}\cdot j$ formula (12) becomes the following:

$f\cdot g=\left(f_{1}+f_{2}\cdot j\right)\cdot\left(g_{1}+g_{2}\cdot
j\right)=Re\left(f\cdot g\right)+Im\left(f\cdot g\right)\cdot j,$ (52)

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 ”real” and “imaginary” parts of the quaternionic
product.

The quaternionic product $\sin p\cdot\cos p.$ Using (30), (31), (32), and (33)
we have in this case the following expressions:

$\displaystyle f_{1}=\Phi_{{1\left(\sin
p\right)}}=\frac{\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2}-\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{4V}},$
$\displaystyle f_{2}=\Phi_{{2\left(\sin
p\right)}}=-\frac{\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{2V}}b,$
$\displaystyle g_{1}=\Phi_{{1\left(\cos
p\right)}}=\frac{\left(e^{{-V}}+e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V}},$
$\displaystyle g_{2}=\Phi_{{2\left(\cos
p\right)}}=\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{2V}}b.$

Substituting these expressions into (52) and taking into account that
$V=\overline{V}$ and $\beta=\overline{\beta}$ we obtain as follows:

$\begin{gathered}Re\left(\sin p\cdot\cos
p\right)=f_{1}g_{1}-f_{2}\overline{g_{2}}\\\
=\left[\frac{\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2}-\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{4V}}\right]\left[\frac{\left(e^{{-V}}+e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2}\right.\\\
+\left.\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V}}\right]+\frac{\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2V}\,b\,\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2V}\,\overline{b},\end{gathered}$
(53) $\begin{gathered}Im\left(\sin p\cdot\cos
p\right)=f_{2}\overline{g_{1}}+f_{1}g_{2}\\\
=-\frac{\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2V}\,b\,\left[\frac{\left(e^{{-V}}+e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V}}\right]\\\
+\left[\frac{\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2}-\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{4V}}\right]\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2V}\,b\\\
=\frac{\left(e^{{-2V}}-e^{2V}\right)}{4V}\,b\,\left(-\cos^{2}\frac{a+\overline{a}}{2}+\sin^{2}\frac{a+\overline{a}}{2}\right).\end{gathered}$
(54)

The quaternionic product $\cos p\cdot\sin p.$ In this case we must substitute
into (52) the following expressions:

$\displaystyle f_{1}$ $\displaystyle=\Phi_{{1\left(\cos
p\right)}}=\frac{\left(e^{{-V}}+e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V}},$
$\displaystyle f_{2}$ $\displaystyle=\Phi_{{2\left(\cos
p\right)}}=\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{2V}}b,$
$\displaystyle g_{1}$ $\displaystyle=\Phi_{{1\left(\sin
p\right)}}=\frac{\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2}-\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{4V}},$
$\displaystyle g_{2}$ $\displaystyle=\Phi_{{2\left(\sin
p\right)}}=-\frac{\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{2V}}b.$

After substitution we have as follows:

$\begin{gathered}Re\left(\cos p\cdot\sin
p\right)=f_{1}g_{1}-f_{2}\overline{g_{2}}\\\
=\left[\frac{\left(e^{{-V}}+e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V}}\right]\left[\frac{\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2}\right.\\\
-\left.\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{4V}}\right]+\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2V}\,b\,\frac{\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2V}\overline{b},\end{gathered}$
(55) $\begin{gathered}Im\left(\cos p\cdot\sin
p\right)=f_{2}\overline{{g_{1}}}+f_{1}g_{2}\\\
=\frac{\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2V}\,b\,\left[\frac{\left(e^{{-V}}+e^{V}\right)\sin\frac{a+\overline{a}}{2}}{2}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{{4V}}\right]\\\
-\left[\frac{\left(e^{{-V}}+e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2}+\frac{\left(a-\overline{a}\right)\left(e^{{-V}}-e^{V}\right)\sin\frac{a+\overline{a}}{2}}{{4V}}\right]\frac{\left(e^{{-V}}-e^{V}\right)\cos\frac{a+\overline{a}}{2}}{2V}\,b\\\
=\frac{\left(e^{{-2V}}-e^{2V}\right)}{4V}\,b\,\left(\sin^{2}\frac{a+\overline{a}}{2}-\cos^{2}\frac{a+\overline{a}}{2}\right).\end{gathered}$
(56)

Comparing (53) and (55) as well as (54) and (56) we see that

$Re\left(\sin p\cdot\cos p\right)=Re\left(\cos p\cdot\sin
p\right),\,\,\,\,\,\,Im\left(\sin p\cdot\cos p\right)=Im\left(\cos p\cdot\sin
p\right).$

Thus we have shown that the quaternionic multiplication of H-holomorphic
functions $\sin p$ and $\cos p$ behaves as commutative.

In conclusion, it is not superfluous to note that the theory of H-holomorphic
functions in question possesses, in a sense, the so-called "internal
perfection", since this theory follows in essence from one main idea [4, p.
15]: "each point of any real line is at the same time a point of some plane
and space as a whole, and therefore any characterization of differentiability
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 leads to introducing the conception of one holistic
notion of a holomorphic function that in the case of complex variables has a
C-representation and in the case of quaternions has a corresponding
H-representation. Moreover, such a single vision is also confirmed by the
studied quaternionic power series, which in C- and H-representations of any
holomorphic function are similar and converge with identical convergence
radiuses. Analogously, decompositions of C-analytic and H-analytic functions
have the same radiuses of convergence. We also can speak of the so-called
"external justification" of the theory in question, since this theory is, as
far as we know, the only theory, which explains the fact that quaternionic
multiplication of H-holomorphic functions behaves as commutative. This is a
reasonable ground to approve that the presented theory of H-holomorphic
functions is true.

## References

  * [1] Sudbery, A., Quaternionic analysis. Math. Proc. Camb. Phil. Soc., 85, pp.199-225, 1979.
  * [2] Centili, G,. Stoppato, C., Power series and analyticity over the quaternions, 2009, avaliable at arXiv: 0902.4679.
  * [3] Gürlebeck, K., Habetha, K., Sprößig, W., Funktionentheorie in der Ebene und im Raum, Birkhäuser Verlag, Basel - Boston – Berlin, 2006.
  * [4] Parfenov, M., Essentially Adequate Concept of Holomorphic Functions in Quaternionic Analysis, American Journal of Mathematical Analysis. vol. 8, No. 1, 2020, pp.14-30. avaliable at ajma-8-1-3.
  * [5] Parfenov, M., On the notion of a quaternionic holomorphic function, 2024, avaliable at arXiv: 2402.08487.
  * [6] Kantor, I. L., Solodovnikov, A. S., Hypercomplex Numbers. An Elementary Introduction to Algebras, Springer-Verlag, New York-Berlin-Heidelberg-London-Paris-Tokyo, 1989, avaliable to download at here.
  * [7] Parfenov, M., On Properties of Holomorphic Functions in Quaternionic Analysis, American Journal of Mathematical Analysis. Vol. 5, No. 1, 2017, pp. 17-24. avaliable at ajma-5-1-4.
  * [8] Mathews, J. H., Howell, R. W., Complex Analysis for Mathematics and Engineering, 3rd ed., Jones and Bartlett Publishers, 1997.
  * [9] Shabat, B.V., An introduction to Complex Analysis, Nauka, 1961, (in Russian), avaliable at here.
  * [10] Polyanin, Andrei D., Manzhirov, Alexander V., Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC, 2007.
  * [11] Parfenov, M., A Quaternionic Potential Conception with Applying to 3D Potential Fields, American Journal of Mathematical Analysis. Vol. 7, No. 1, 2019, pp. 1-10. avaliable at ajma-7-1-1.
  * [12] Apostol, Tom M., Mathematical Analysis, Second Edition, Pearson Education, Inc., 1974.