# The Regularity of Functions on Dual Split Quaternions in Clifford Analysis

Authors: Ji Eun Kim; Kwang Ho Shon
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions and provides differential operators in dual split quaternions and a dual split regular function onΩ
⊂
ℂ
2
×
ℂ
2 that has a dual split Cauchy-Riemann system in dual split quaternions.


---

## Body

## 1. Introduction

Hamilton introduced quaternions, extending complex numbers to higher spatial dimensions in differential geometry (see [1]). A set of quaternions can be represented as
(1)
ℋ
=
{
z
=
x
0
+
x
1
i
+
x
2
j
+
x
3
k
:
x
m
∈
ℝ
,
m
=
0,1
,
2,3
}
,
where i
2
=
j
2
=
k
2
=
-
1, i
j
k
=
-
1, and ℝ denotes the set of real numbers. Cockle [2] introduced a set of split quaternions as
(2)
𝒮
=
{
z
=
x
0
+
x
1
e
1
+
x
2
e
2
+
x
3
e
3
:
x
m
∈
ℝ
,
m
=
0,1
,
2,3
}
,
where e
1
2
=
-
1, e
2
2
=
e
3
2
=
1, and e
1
e
2
e
3
=
1. A set of split quaternions is noncommutative and contains zero divisors, nilpotent elements, and nontrivial idempotents (see [3, 4]). Previous studies have examined the geometric and physical applications of split quaternions, which are required in solving split quaternionic equations (see [5, 6]). Inoguchi [7] reformulated the Gauss-Codazzi equations in forms consistent with the theory of integrable systems in the Minkowski 3-space for split quaternion numbers.A dual quaternion can be represented in a form reflecting an ordinary quaternion and a dual symbol. Because dual-quaternion algebra is constructed from real eight-dimensional vector spaces and an ordered pair of quaternions, dual quaternions are used in computer vision applications. Kenwright [8] provided the characteristics of dual quaternions, and Pennestrì and Stefanelli [9] examined some properties by using dual quaternions. Son [10, 11] offered an extension problem for solutions of partial differential equations and generalized solutions for the Riesz system. By using properties of Hamilton operators, Kula and Yayli [4] defined dual split quaternions and gave some properties of the screw motion in the Minkowski 3-space, showing that ℋ has a rotation with unit split quaternions in ℋ and a scalar product that allows it to be identified with the semi-Euclidean space for split quaternion numbers.It was shown (see [12, 13]) that any complex-valued harmonic function f
1 in a pseudoconvex domain D of ℂ
2
×
ℂ
2, ℂ being the set of complex numbers, has a conjugate function f
2 in D such that the quaternion-valued function f
1
+
f
2
j is hyperholomorphic in D and gave a regeneration theorem in a quaternion analysis in view of complex and Clifford analysis. In addition, we [14, 15] provided a new expression of the quaternionic basis and a regular function on reduced quaternions by associating hypercomplex numbers e
1 and e
2. We [16] investigated the existence of hyperconjugate harmonic functions of an octonion number system, and we [17, 18] obtained some regular functions with values in dual quaternions and researched an extension problem for properties of regular functions with values in dual quaternions and some applications for such problems.This paper provides a regular function and some properties of differential operators in dual split quaternions. In addition, we research some equivalent conditions for Cauchy-Riemann systems and expressions of power series in dual split quaternions from the definition of dual split regular on an open setΩ
⊂
ℂ
2
×
ℂ
2.


## 2. Preliminaries

A dual numberA has the form a
+
ε
b, where a and b are real numbers and ε is a dual symbol subject to the rules
(3)
ε
≠
0
,
0
ε
=
ε
0
=
0
,
1
ε
=
ε
1
=
ε
,
ε
2
=
0
,
and a split quaternion q
∈
𝒮 is an expression of the form
(4)
q
=
x
0
+
x
1
e
1
+
x
2
e
2
+
x
3
e
3
,
where x
m
∈
ℝ
(
m
=
0,1
,
2,3
) and e
r
(
r
=
1,2
,
3
) are split quaternionic units satisfying noncommutative multiplication rules (for split quaternions, see [1]):
(5)
e
1
2
=
-
1
,
e
2
2
=
e
3
2
=
1
,
e
1
e
2
=
-
e
2
e
1
=
e
3
,
e
2
e
3
=
-
e
3
e
2
=
-
e
1
,
e
3
e
1
=
-
e
1
e
3
=
e
2
.
Similarly, a dual split quaternion z can be written as
(6)
𝒟
(
𝒮
)
=
{
z
∣
z
=
p
0
+
ε
p
1
,
p
r
∈
𝒮
,
r
=
0,1
}
,
which has elements of the following form:
(7)
z
=
{
(
x
0
+
x
1
e
1
)
+
(
x
2
+
x
3
e
1
)
e
2
}
+
ε
{
(
y
0
+
y
1
e
1
)
+
(
y
2
+
y
3
e
1
)
e
2
}
=
(
z
0
+
z
1
e
2
)
+
ε
(
z
2
+
z
3
e
2
)
=
p
0
+
ε
p
1
,
where p
0
=
z
0
+
z
1
e
2 and p
1
=
z
2
+
z
3
e
2 are split quaternion components, z
0
=
x
0
+
x
1
e
1, z
1
=
x
2
+
x
3
e
1, z
2
=
y
0
+
y
1
e
1, and z
3
=
y
2
+
y
3
e
1 are usual complex numbers, and x
m
,
y
m
∈
ℝ
(
m
=
0,1
,
2,3
). The multiplication of split quaternionic units with a dual symbol is commutative ε
e
r
=
e
r
ε
(
r
=
1,2
,
3
). However, by properties of split quaternionic unit,
(8)
z
k
e
r
=
e
r
z
k
(
k
=
0,1
,
2,3
,
r
=
0,1
)
,
z
k
e
r
=
e
r
z
k
¯
(
k
=
0,1
,
2,3
,
r
=
2,3
)
,
e
r
p
k
≠
p
k
e
r
,
e
r
p
k
=
p
(
k
r
)
e
r
(
r
=
1,2
,
3
,
k
=
0,1
)
,
where
(9)
p
(
01
)
=
z
0
-
z
1
e
2
=
x
0
+
x
1
e
1
-
x
2
e
2
-
x
3
e
3
,
p
(
02
)
=
z
0
¯
+
z
1
¯
e
2
=
x
0
-
x
1
e
1
+
x
2
e
2
-
x
3
e
3
,
p
(
03
)
=
z
0
¯
-
z
1
¯
e
2
=
x
0
-
x
1
e
1
-
x
2
e
2
+
x
3
e
3
,
p
(
11
)
=
z
2
-
z
3
e
2
=
y
0
+
y
1
e
1
-
y
2
e
2
-
y
3
e
3
,
p
(
12
)
=
z
2
¯
+
z
3
¯
e
2
=
y
0
-
y
1
e
1
+
y
2
e
2
-
y
3
e
3
,
p
(
13
)
=
z
2
¯
-
z
3
¯
e
2
=
y
0
-
y
1
e
1
-
y
2
e
2
+
y
3
e
3
,
with z
0
¯
=
x
0
-
x
1
e
1, z
1
¯
=
x
2
-
x
3
e
1, z
2
¯
=
y
0
-
y
1
e
1, and z
3
¯
=
y
2
-
y
3
e
1. For instance,
(10)
e
2
p
0
=
e
2
(
x
0
+
x
1
e
1
+
x
2
e
2
+
x
3
e
3
)
=
(
x
0
-
x
1
e
1
+
x
2
e
2
-
x
3
e
3
)
e
2
=
p
(
02
)
e
2
,
e
1
p
1
=
e
1
(
y
0
+
y
1
e
1
+
y
2
e
2
+
y
3
e
3
)
=
(
y
0
+
y
1
e
1
-
y
2
e
2
-
y
3
e
3
)
e
1
=
p
(
11
)
e
1
.
Because of the properties of the eight-unit equality, the addition and subtraction of dual split quaternions are governed by the rules of ordinary algebra. Here the symbol p
(
k
r
) is used by just enumerating r and k, not r times k. For example, p
(
22
)
≠
p
4 and p
22
=
p
4.For any two elementsz
=
p
0
+
ε
p
1 and w
=
q
0
+
ε
q
1 of 𝒟
(
𝒮
), where q
0
=
∑
r
=
0
3
‍
s
r
e
r and q
1
=
∑
r
=
0
3
‍
t
r
e
r are split quaternion components and s
r
,
t
r
∈
ℝ
(
r
=
0,1
,
2,3
), their noncommutative product is given by
(11)
z
w
=
(
p
0
+
ε
p
1
)
(
q
0
+
ε
q
1
)
=
p
0
q
0
+
ε
(
p
0
q
1
+
p
1
q
0
)
.
The conjugation z
* of z and the corresponding modulus z
z
* in 𝒟
(
𝒮
) are defined by
(12)
z
*
=
p
0
*
+
ε
p
1
*
,
z
z
*
=
z
*
z
=
p
0
p
0
*
+
ε
(
p
0
p
1
*
+
p
1
p
0
*
)
z
z
*
=
(
z
0
z
0
¯
-
z
1
z
1
¯
)
+
2
ε
(
z
0
z
2
¯
-
z
1
z
3
¯
)
z
z
*
=
∑
r
=
0
1
‍
{
(
x
r
2
-
x
r
+
2
2
)
+
ε
(
x
r
y
r
-
x
r
+
2
y
r
+
2
)
}
,
where p
0
*
=
z
0
¯
-
z
1
e
2 and p
1
*
=
z
2
¯
-
z
3
e
2.Lemma 1.
For allz
∈
𝒟
(
𝒮
) and n
∈
ℕ
:
=
{
1,2
,
3
,
…
}, we have
(13)
z
n
=
p
0
n
+
ε
∑
k
=
1
n
‍
p
0
n
-
k
p
1
p
0
k
-
1
.Proof.
Ifn
=
1, then (13) is trivial. Now suppose that this holds for some n
∈
ℕ. Then, as desired,
(14)
z
n
+
1
=
z
z
n
=
z
(
p
0
n
+
ε
∑
k
=
1
n
‍
p
0
n
-
k
p
1
p
0
k
-
1
)
=
p
0
n
+
1
+
ε
∑
k
=
1
n
‍
p
0
n
-
k
+
1
p
1
p
0
k
-
1
+
ε
p
1
p
0
n
=
p
0
n
+
1
+
ε
∑
k
=
1
n
+
1
‍
p
0
n
+
1
-
k
p
1
p
0
k
-
1
.
By the principle of mathematical induction, (13) holds for all n
∈
ℕ.LetΩ be an open subset of ℂ
2
×
ℂ
2. Then the function f
:
Ω
→
𝒟
(
𝒮
) can be expressed as
(15)
f
(
z
)
=
f
(
p
0
,
p
1
)
=
f
0
(
p
0
,
p
1
)
+
ε
f
1
(
p
0
,
p
1
)
,
where the component functions f
r
:
Ω
→
𝒮
(
r
=
0,1
) are split quaternionic-valued functions. The component functions f
r
(
r
=
0,1
) are
(16)
f
0
(
p
0
,
p
1
)
=
f
0
(
z
0
,
z
1
,
z
2
,
z
3
)
=
g
0
(
z
0
,
z
1
,
z
2
,
z
3
)
+
g
1
(
z
0
,
z
1
,
z
2
,
z
3
)
e
2
,
f
1
(
p
0
,
p
1
)
=
f
1
(
z
0
,
z
1
,
z
2
,
z
3
)
=
g
2
(
z
0
,
z
1
,
z
2
,
z
3
)
+
g
3
(
z
0
,
z
1
,
z
2
,
z
3
)
e
2
,
where g
k
=
u
2
k
+
u
2
k
+
1
e
1
(
k
=
0,1
) and g
k
=
v
2
k
-
4
+
v
2
k
-
3
e
1
(
k
=
2,3
) are complex-valued functions, and u
r and v
r
(
r
=
0,1
,
2,3
) are real-valued functions.Now, we let differential operatorsD
1 and D
2 be defined on 𝒟
(
𝒮
) as
(17)
D
1
:
=
D
(
11
)
+
ε
D
(
12
)
,
D
2
:
=
D
(
21
)
+
ε
D
(
22
)
.
Then the conjugate operators D
1
* and D
2
* are
(18)
D
1
*
=
D
(
11
)
*
+
ε
D
(
12
)
*
,
D
2
*
=
D
(
21
)
*
+
ε
D
(
22
)
*
,
where
(19)
D
(
11
)
=
∂
∂
z
0
+
∂
∂
z
1
¯
e
2
=
1
2
(
∂
∂
x
0
-
∂
∂
x
1
e
1
+
∂
∂
x
2
e
2
+
∂
∂
x
3
e
3
)
,
D
(
12
)
=
∂
∂
z
2
+
∂
∂
z
3
¯
e
2
=
1
2
(
∂
∂
y
0
-
∂
∂
y
1
e
1
+
∂
∂
y
2
e
2
+
∂
∂
y
3
e
3
)
,
D
(
21
)
=
∂
∂
z
0
+
1
2
∂
∂
z
1
e
2
=
1
2
(
∂
∂
x
0
-
∂
∂
x
1
e
1
+
1
2
∂
∂
x
2
e
2
-
1
2
∂
∂
x
3
e
3
)
,
D
(
22
)
=
∂
∂
z
2
+
1
2
∂
∂
z
3
e
2
=
1
2
(
∂
∂
y
0
-
∂
∂
y
1
e
1
+
1
2
∂
∂
y
2
e
2
-
1
2
∂
∂
y
3
e
3
)
,
(20)
D
(
11
)
*
=
∂
∂
z
0
¯
-
∂
∂
z
1
¯
e
2
=
1
2
(
∂
∂
x
0
+
∂
∂
x
1
e
1
-
∂
∂
x
2
e
2
-
∂
∂
x
3
e
3
)
,
D
(
12
)
*
=
∂
∂
z
2
¯
-
∂
∂
z
3
¯
e
2
=
1
2
(
∂
∂
y
0
+
∂
∂
y
1
e
1
-
∂
∂
y
2
e
2
-
∂
∂
y
3
e
3
)
,
D
(
21
)
*
=
∂
∂
z
0
¯
-
1
2
∂
∂
z
1
e
2
=
1
2
(
∂
∂
x
0
+
∂
∂
x
1
e
1
-
1
2
∂
∂
x
2
e
2
+
1
2
∂
∂
x
3
e
3
)
,
D
(
22
)
*
=
∂
∂
z
2
¯
-
1
2
∂
∂
z
3
e
2
=
1
2
(
∂
∂
y
0
+
∂
∂
y
1
e
1
-
1
2
∂
∂
y
2
e
2
+
1
2
∂
∂
y
3
e
3
)
act on 𝒟
(
𝒮
). These operators are called corresponding Cauchy-Riemann operators in 𝒟
(
𝒮
), where ∂
/
∂
z
r and ∂
/
∂
z
r
¯
(
r
=
0,1
,
2,3
) are usual differential operators used in the complex analysis.Remark 2.
From the definition of differential operators on𝒟
(
𝒮
),
(21)
D
r
f
=
(
D
(
r
1
)
+
ε
D
(
r
2
)
)
(
f
0
+
ε
f
1
)
=
D
(
r
1
)
f
0
+
ε
(
D
(
r
1
)
f
1
+
D
(
r
2
)
f
0
)
,
D
r
*
f
=
(
D
(
r
1
)
*
+
ε
D
(
r
2
)
*
)
(
f
0
+
ε
f
1
)
=
D
(
r
1
)
*
f
0
+
ε
(
D
(
r
1
)
*
f
1
+
D
(
r
2
)
*
f
0
)
,
where r
=
1,2.Definition 3.
LetΩ be an open set in ℂ
2
×
ℂ
2. A function f
=
f
0
+
ε
f
1 is called an L
r (resp., R
r)-regular function (
r
=
1,2
) on Ω if the following two conditions are satisfied:(i)
f
k
(
k
=
0,1
) are continuously differential functions on Ω, and
(ii)
D
r
*
f
(
z
)
=
0 (resp., f
(
z
)
D
r
*
=
0) on Ω
(
r
=
1,2
).In particular, the equationD
1
*
f
(
z
)
=
0 of Definition 3 is equivalent to
(22)
D
(
11
)
*
f
0
=
0
,
D
(
12
)
*
f
0
+
D
(
11
)
*
f
1
=
0
.
In addition,
(23)
∂
g
0
∂
z
0
¯
-
∂
g
1
¯
∂
z
1
¯
=
0
,
∂
g
1
∂
z
0
¯
-
∂
g
0
¯
∂
z
1
¯
=
0
,
∂
g
2
∂
z
0
¯
+
∂
g
0
∂
z
2
¯
-
∂
g
3
¯
∂
z
1
¯
-
∂
g
1
¯
∂
z
3
¯
=
0
,
∂
g
3
∂
z
0
¯
+
∂
g
1
∂
z
2
¯
-
∂
g
2
¯
∂
z
1
¯
-
∂
g
0
¯
∂
z
3
¯
=
0
.
Concretely, the following system is obtained:
(24)
∂
u
0
∂
x
0
-
∂
u
1
∂
x
1
-
∂
u
2
∂
x
2
-
∂
u
3
∂
x
3
=
0
,
∂
u
1
∂
x
0
+
∂
u
0
∂
x
1
-
∂
u
2
∂
x
3
+
∂
u
3
∂
x
2
=
0
,
∂
u
2
∂
x
0
-
∂
u
3
∂
x
1
-
∂
u
0
∂
x
2
-
∂
u
1
∂
x
3
=
0
,
∂
u
3
∂
x
0
+
∂
u
2
∂
x
1
-
∂
u
0
∂
x
3
+
∂
u
1
∂
x
2
=
0
,
∂
u
0
∂
y
0
-
∂
u
1
∂
y
1
-
∂
u
2
∂
y
2
-
∂
u
3
∂
y
3
+
∂
v
0
∂
x
0
-
∂
v
1
∂
x
1
-
∂
v
2
∂
x
2
-
∂
v
3
∂
x
3
=
0
,
∂
u
1
∂
y
0
+
∂
u
0
∂
y
1
-
∂
u
2
∂
y
3
+
∂
u
3
∂
y
2
+
∂
v
1
∂
x
0
+
∂
v
0
∂
x
1
-
∂
v
2
∂
x
3
+
∂
v
3
∂
x
2
=
0
,
∂
u
2
∂
y
0
-
∂
u
3
∂
y
1
-
∂
u
0
∂
y
2
-
∂
u
1
∂
y
3
+
∂
v
2
∂
x
0
-
∂
v
3
∂
x
1
-
∂
v
0
∂
x
2
-
∂
v
1
∂
x
3
=
0
,
∂
u
3
∂
y
0
+
∂
u
2
∂
y
1
-
∂
u
0
∂
y
3
+
∂
u
1
∂
y
2
+
∂
v
3
∂
x
0
+
∂
v
2
∂
x
1
-
∂
v
0
∂
x
3
+
∂
v
1
∂
x
2
=
0
.
The above systems (23) and (24) are corresponding Cauchy-Riemann systems in 𝒟
(
𝒮
). Similarly, the equation D
2
*
f
(
z
)
=
0 of Definition 3 is equivalent to
(25)
D
(
21
)
*
f
0
=
0
,
D
(
22
)
*
f
0
+
D
(
21
)
*
f
1
=
0
.
Then,
(26)
∂
g
0
∂
z
0
¯
-
1
2
∂
g
1
¯
∂
z
1
=
0
,
∂
g
1
∂
z
0
¯
-
1
2
∂
g
0
¯
∂
z
1
=
0
,
∂
g
2
∂
z
0
¯
+
∂
g
0
∂
z
2
¯
-
1
2
∂
g
3
¯
∂
z
1
-
1
2
∂
g
1
¯
∂
z
3
=
0
,
∂
g
3
∂
z
0
¯
+
∂
g
1
∂
z
2
¯
-
1
2
∂
g
2
¯
∂
z
1
-
1
2
∂
g
0
¯
∂
z
3
=
0
.
Concretely, the following system is obtained:
(27)
∂
u
0
∂
x
0
-
∂
u
1
∂
x
1
-
1
2
∂
u
2
∂
x
2
+
1
2
∂
u
3
∂
x
3
=
0
,
∂
u
1
∂
x
0
+
∂
u
0
∂
x
1
+
1
2
∂
u
2
∂
x
3
+
1
2
∂
u
3
∂
x
2
=
0
,
∂
u
2
∂
x
0
-
∂
u
3
∂
x
1
-
1
2
∂
u
0
∂
x
2
+
1
2
∂
u
1
∂
x
3
=
0
,
∂
u
3
∂
x
0
+
∂
u
2
∂
x
1
+
1
2
∂
u
0
∂
x
3
+
1
2
∂
u
1
∂
x
2
=
0
,
∂
u
0
∂
y
0
-
∂
u
1
∂
y
1
-
1
2
∂
u
2
∂
y
2
+
1
2
∂
u
3
∂
y
3
+
∂
v
0
∂
x
0
-
∂
v
1
∂
x
1
-
1
2
∂
v
2
∂
x
2
+
1
2
∂
v
3
∂
x
3
=
0
,
∂
u
1
∂
y
0
+
∂
u
0
∂
y
1
+
1
2
∂
u
2
∂
y
3
+
1
2
∂
u
3
∂
y
2
+
∂
v
1
∂
x
0
+
∂
v
0
∂
x
1
+
1
2
∂
v
2
∂
x
3
+
1
2
∂
v
3
∂
x
2
=
0
,
∂
u
2
∂
y
0
-
∂
u
3
∂
y
1
-
1
2
∂
u
0
∂
y
2
+
1
2
∂
u
1
∂
y
3
+
∂
v
2
∂
x
0
-
∂
v
3
∂
x
1
-
1
2
∂
v
0
∂
x
2
+
1
2
∂
v
1
∂
x
3
=
0
,
∂
u
3
∂
y
0
+
∂
u
2
∂
y
1
+
1
2
∂
u
0
∂
y
3
+
1
2
∂
u
1
∂
y
2
+
∂
v
3
∂
x
0
+
∂
v
2
∂
x
1
+
1
2
∂
v
0
∂
x
3
+
1
2
∂
v
1
∂
x
2
=
0
.
The above systems (26) and (27) are corresponding Cauchy-Riemann systems in 𝒟
(
𝒮
).On the other hand, the equationf
(
z
)
D
1
*
=
0 of Definition 3 is equivalent to
(28)
f
0
D
(
11
)
*
=
0
,
f
0
D
(
12
)
*
=
-
f
1
D
(
11
)
*
.
Then,
(29)
g
0
∂
∂
z
0
¯
=
g
1
∂
∂
z
1
,
g
1
∂
∂
z
0
=
g
0
∂
∂
z
1
¯
,
g
0
∂
∂
z
2
¯
-
g
1
∂
∂
z
3
=
-
g
2
∂
∂
z
0
¯
+
g
3
∂
∂
z
1
,
g
1
∂
∂
z
2
-
g
0
∂
∂
z
3
¯
=
-
g
3
∂
∂
z
0
+
g
2
∂
∂
z
1
¯
,
where
(30)
g
k
∂
∂
z
m
¯
=
∂
g
k
∂
z
m
¯
,
g
k
∂
∂
z
m
=
∂
g
k
∂
z
m
(
k
,
m
=
0,1
,
2,3
)
.
Concretely, the following system is obtained:
(31)
∂
u
0
∂
x
0
-
∂
u
1
∂
x
1
=
∂
u
2
∂
x
2
+
∂
u
3
∂
x
3
,
∂
u
1
∂
x
0
+
∂
u
0
∂
x
1
=
-
∂
u
2
∂
x
3
+
∂
u
3
∂
x
2
,
∂
u
2
∂
x
0
+
∂
u
3
∂
x
1
=
∂
u
0
∂
x
2
-
∂
u
1
∂
x
3
,
∂
u
3
∂
x
0
-
∂
u
2
∂
x
1
=
∂
u
0
∂
x
3
+
∂
u
1
∂
x
2
,
∂
u
0
∂
y
0
-
∂
u
1
∂
y
1
-
∂
u
2
∂
y
2
-
∂
u
3
∂
y
3
=
-
∂
v
0
∂
x
0
+
∂
v
1
∂
x
1
+
∂
v
2
∂
x
2
+
∂
v
3
∂
x
3
,
∂
u
1
∂
y
0
+
∂
u
0
∂
y
1
+
∂
u
2
∂
y
3
-
∂
u
3
∂
y
2
=
-
∂
v
1
∂
x
0
-
∂
v
0
∂
x
1
-
∂
v
2
∂
x
3
+
∂
v
3
∂
x
2
,
∂
u
2
∂
y
0
+
∂
u
3
∂
y
1
-
∂
u
0
∂
y
2
+
∂
u
1
∂
y
3
=
-
∂
v
2
∂
x
0
-
∂
v
3
∂
x
1
+
∂
v
0
∂
x
2
-
∂
v
1
∂
x
3
,
∂
u
3
∂
y
0
-
∂
u
2
∂
y
1
-
∂
u
0
∂
y
3
-
∂
u
1
∂
y
2
=
-
∂
v
3
∂
x
0
+
∂
v
2
∂
x
1
+
∂
v
0
∂
x
3
+
∂
v
1
∂
x
2
.
Similarly, the equation f
(
z
)
D
2
*
=
0 of Definition 3 is equivalent to
(32)
f
0
D
(
21
)
*
=
0
,
f
0
D
(
22
)
*
=
-
f
1
D
(
21
)
*
.
Then,
(33)
g
0
∂
∂
z
0
¯
=
1
2
g
1
∂
∂
z
1
¯
,
g
1
∂
∂
z
0
=
1
2
g
0
∂
∂
z
1
,
g
0
∂
∂
z
2
¯
-
1
2
g
1
∂
∂
z
3
¯
=
-
g
2
∂
∂
z
0
¯
+
1
2
g
3
∂
∂
z
1
¯
,
g
1
∂
∂
z
2
-
1
2
g
0
∂
∂
z
3
=
-
g
3
∂
∂
z
0
+
1
2
g
2
∂
∂
z
1
,
where
(34)
g
k
∂
∂
z
m
¯
=
∂
g
k
∂
z
m
¯
,
g
k
∂
∂
z
m
=
∂
g
k
∂
z
m
(
k
,
m
=
0,1
,
2,3
)
.
Concretely, the system is obtained as follows:
(35)
∂
u
0
∂
x
0
-
∂
u
1
∂
x
1
-
1
2
∂
u
2
∂
x
2
+
1
2
∂
u
3
∂
x
3
=
0
,
∂
u
1
∂
x
0
+
∂
u
0
∂
x
1
-
1
2
∂
u
2
∂
x
3
-
1
2
∂
u
3
∂
x
2
=
0
,
∂
u
2
∂
x
0
+
∂
u
3
∂
x
1
-
1
2
∂
u
0
∂
x
2
-
1
2
∂
u
1
∂
x
3
=
0
,
∂
u
3
∂
x
0
-
∂
u
2
∂
x
1
+
1
2
∂
u
0
∂
x
3
-
1
2
∂
u
1
∂
x
2
=
0
,
∂
u
0
∂
y
0
-
∂
u
1
∂
y
1
-
1
2
∂
u
2
∂
y
2
+
1
2
∂
u
3
∂
y
3
+
∂
v
0
∂
x
0
∂
u
0
∂
y
0
-
∂
v
1
∂
x
1
-
1
2
∂
v
2
∂
x
2
+
1
2
∂
v
3
∂
x
3
=
0
,
∂
u
1
∂
y
0
+
∂
u
0
∂
y
1
-
1
2
∂
u
2
∂
y
3
-
1
2
∂
u
3
∂
y
2
+
∂
v
1
∂
x
0
∂
u
0
∂
y
0
+
∂
v
0
∂
x
1
-
1
2
∂
v
2
∂
x
3
-
1
2
∂
v
3
∂
x
2
=
0
,
∂
u
2
∂
y
0
+
∂
u
3
∂
y
1
-
1
2
∂
u
0
∂
y
2
+
1
2
∂
u
1
∂
y
3
+
∂
v
2
∂
x
0
∂
u
0
∂
y
0
+
∂
v
3
∂
x
1
-
1
2
∂
v
0
∂
x
2
+
1
2
∂
v
1
∂
x
3
=
0
,
∂
u
3
∂
y
0
-
∂
u
2
∂
y
1
-
1
2
∂
u
0
∂
y
3
-
1
2
∂
u
1
∂
y
2
+
∂
v
3
∂
x
0
∂
u
0
∂
y
0
-
∂
v
2
∂
x
1
-
1
2
∂
v
0
∂
x
3
-
1
2
∂
v
1
∂
x
2
=
0
.
From the systems (24), (27), (31), and (35), the equations D
r
*
f
(
z
)
=
0 and f
(
z
)
D
r
*
=
0
(
r
=
1,2
) are different. Therefore, the equations D
r
*
f
(
z
)
=
0 and f
(
z
)
D
r
*
=
0
(
r
=
1,2
) should be distinguished as L
r-regular functions (
r
=
1,2
) and R
r-regular functions (
r
=
1,2
) on Ω, respectively. Now the properties of the L
r-regular function (
r
=
1,2
) with values in 𝒟
(
𝒮
) are considered.


## 3. Properties ofL
r-Regular Functions (
r
=
1,2
) with Values in 𝒟
(
𝒮
)

We consider properties of aL
r-regular functions (
r
=
1,2
) with values in 𝒟
(
𝒮
).Theorem 4.
LetΩ be an open set in ℂ
2
×
ℂ
2 and let f
=
f
0
+
ε
f
1
=
(
g
0
+
g
1
e
2
)
+
ε
(
g
2
+
g
3
e
2
) be an L
1-regular function defined on Ω. Then
(36)
D
1
f
=
{
2
(
∂
∂
z
1
¯
+
ε
∂
∂
z
3
¯
)
e
2
-
(
∂
∂
x
1
+
ε
∂
∂
y
1
)
e
1
}
f
.Proof.
By the system (23), we have
(37)
D
1
f
=
D
(
11
)
f
0
+
ε
(
D
(
12
)
f
0
+
D
(
11
)
f
1
)
=
(
∂
g
0
∂
z
0
+
∂
g
1
¯
∂
z
1
¯
)
+
(
∂
g
1
∂
z
0
+
∂
g
0
¯
∂
z
1
¯
)
e
2
+
ε
(
∂
g
0
∂
z
2
+
∂
g
1
¯
∂
z
3
¯
+
∂
g
2
∂
z
0
+
∂
g
3
¯
∂
z
1
¯
)
+
ε
(
∂
g
1
∂
z
2
+
∂
g
0
¯
∂
z
3
¯
+
∂
g
3
∂
z
0
+
∂
g
2
¯
∂
z
1
¯
)
e
2
=
(
∂
g
0
∂
z
0
¯
+
∂
u
1
∂
x
1
-
∂
u
0
∂
x
1
e
1
+
∂
g
1
¯
∂
z
1
¯
)
+
(
∂
g
1
∂
z
0
¯
+
∂
u
3
∂
x
1
-
∂
u
2
∂
x
1
e
1
+
∂
g
0
¯
∂
z
1
¯
)
e
2
+
ε
(
∂
g
0
∂
z
2
¯
+
∂
u
1
∂
y
1
-
∂
u
0
∂
y
1
e
1
+
∂
g
1
¯
∂
z
3
¯
+
∂
g
2
∂
z
0
¯
+
∂
v
1
∂
x
1
-
∂
v
0
∂
x
1
e
1
+
∂
g
3
¯
∂
z
1
¯
)
+
ε
(
∂
g
1
∂
z
2
¯
+
∂
u
3
∂
y
1
-
∂
u
2
∂
y
1
e
1
+
∂
g
0
¯
∂
z
3
¯
+
∂
g
3
∂
z
0
¯
+
∂
v
3
∂
x
1
-
∂
v
2
∂
x
1
e
1
+
∂
g
2
¯
∂
z
1
¯
)
e
2
=
(
∂
u
1
∂
x
1
-
∂
u
0
∂
x
1
e
1
+
2
∂
g
1
¯
∂
z
1
¯
)
+
(
∂
u
3
∂
x
1
-
∂
u
2
∂
x
1
e
1
+
2
∂
g
0
¯
∂
z
1
¯
)
e
2
+
ε
(
∂
u
1
∂
y
1
-
∂
u
0
∂
y
1
e
1
+
2
∂
g
1
¯
∂
z
3
¯
+
∂
v
1
∂
x
1
-
∂
v
0
∂
x
1
e
1
+
2
∂
g
3
¯
∂
z
1
¯
)
+
ε
(
∂
u
3
∂
y
1
-
∂
u
2
∂
y
1
e
1
+
2
∂
g
0
¯
∂
z
3
¯
+
∂
v
3
∂
x
1
-
∂
v
2
∂
x
1
e
1
+
2
∂
g
2
¯
∂
z
1
¯
)
e
2
=
{
2
(
∂
∂
z
1
¯
+
ε
∂
∂
z
3
¯
)
e
2
-
(
∂
∂
x
1
+
ε
∂
∂
y
1
)
e
1
}
f
.
Therefore, we obtain
(38)
D
1
f
=
{
2
(
∂
∂
z
1
¯
+
ε
∂
∂
z
3
¯
)
e
2
-
(
∂
∂
x
1
+
ε
∂
∂
y
1
)
e
1
}
f
.Theorem 5.
LetΩ be an open set in ℂ
2
×
ℂ
2 and f
=
f
0
+
ε
f
1
=
(
g
0
+
g
1
e
2
)
+
ε
(
g
2
+
g
3
e
2
) be an L
2-regular function defined on Ω. Then
(39)
D
2
f
=
{
(
∂
∂
z
1
+
ε
∂
∂
z
3
)
e
2
-
(
∂
∂
x
1
+
ε
∂
∂
y
1
)
e
1
}
f
.Proof.
By the system (26), we have
(40)
D
2
f
=
D
(
21
)
f
0
+
ε
(
D
(
22
)
f
0
+
D
(
21
)
f
1
)
=
(
∂
g
0
∂
z
0
+
1
2
∂
g
1
¯
∂
z
1
)
+
(
∂
g
1
∂
z
0
+
1
2
∂
g
0
¯
∂
z
1
)
e
2
+
ε
(
∂
g
0
∂
z
2
+
1
2
∂
g
1
¯
∂
z
3
+
∂
g
2
∂
z
0
+
1
2
∂
g
3
¯
∂
z
1
)
+
ε
(
∂
g
1
∂
z
2
+
1
2
∂
g
0
¯
∂
z
3
+
∂
g
3
∂
z
0
+
1
2
∂
g
2
¯
∂
z
1
)
e
2
=
(
∂
g
0
∂
z
0
¯
+
∂
u
1
∂
x
1
-
∂
u
0
∂
x
1
e
1
+
1
2
∂
g
1
¯
∂
z
1
)
+
(
∂
g
1
∂
z
0
¯
+
∂
u
3
∂
x
1
-
∂
u
2
∂
x
1
e
1
+
1
2
∂
g
0
¯
∂
z
1
)
e
2
+
ε
(
∂
g
0
∂
z
2
¯
+
∂
u
1
∂
y
1
-
∂
u
0
∂
y
1
e
1
+
1
2
∂
g
1
¯
∂
z
3
+
∂
g
2
∂
z
0
¯
+
∂
v
1
∂
x
1
-
∂
v
0
∂
x
1
e
1
+
1
2
∂
g
3
¯
∂
z
1
)
+
ε
(
∂
g
1
∂
z
2
¯
+
∂
u
3
∂
y
1
-
∂
u
2
∂
y
1
e
1
+
1
2
∂
g
0
¯
∂
z
3
+
∂
g
3
∂
z
0
¯
+
∂
v
3
∂
x
1
-
∂
v
2
∂
x
1
e
1
+
1
2
∂
g
2
¯
∂
z
1
)
e
2
=
(
∂
u
1
∂
x
1
-
∂
u
0
∂
x
1
e
1
+
∂
g
1
¯
∂
z
1
)
+
(
∂
u
3
∂
x
1
-
∂
u
2
∂
x
1
e
1
+
∂
g
0
¯
∂
z
1
)
e
2
+
ε
(
∂
u
1
∂
y
1
-
∂
u
0
∂
y
1
e
1
+
∂
g
1
¯
∂
z
3
+
∂
v
1
∂
x
1
-
∂
v
0
∂
x
1
e
1
+
∂
g
3
¯
∂
z
1
)
+
ε
(
∂
u
3
∂
y
1
-
∂
u
2
∂
y
1
e
1
+
∂
g
0
¯
∂
z
3
+
∂
v
3
∂
x
1
-
∂
v
2
∂
x
1
e
1
+
∂
g
2
¯
∂
z
1
)
e
2
=
{
(
∂
∂
z
1
+
ε
∂
∂
z
3
)
e
2
-
(
∂
∂
x
1
+
ε
∂
∂
y
1
)
e
1
}
f
.
Therefore, we obtain the following equation:
(41)
D
2
f
=
{
(
∂
∂
z
1
+
ε
∂
∂
z
3
)
e
2
-
(
∂
∂
x
1
+
ε
∂
∂
y
1
)
e
1
}
f
.Proposition 6.
From properties of differential operators, the following equations are obtained:(42)
D
(
1
r
)
p
r
-
1
=
2
,
D
(
2
r
)
p
r
-
1
=
1
,
D
(
1
r
)
*
p
r
-
1
=
-
1
,
D
(
2
r
)
*
p
r
-
1
=
0
,
D
(
1
r
)
*
p
r
-
1
*
=
2
,
D
(
2
r
)
*
p
r
-
1
*
=
1
,
D
(
r
1
)
p
1
=
D
(
r
1
)
*
p
1
=
D
(
r
1
)
p
1
*
=
D
(
r
1
)
*
p
1
*
D
(
r
1
)
p
1
=
D
(
r
2
)
p
0
=
D
(
r
2
)
*
p
0
=
D
(
r
2
)
p
0
*
D
(
r
1
)
p
1
=
D
(
r
2
)
*
p
0
*
=
0
(
r
=
1,2
)
.Proof.
By properties of the power of dual split quaternions and derivatives on𝒟
(
𝒮
), the following derivatives are obtained:
(43)
D
(
11
)
p
0
=
1
2
(
∂
∂
x
0
-
∂
∂
x
1
e
1
+
∂
∂
x
2
e
2
+
∂
∂
x
3
e
3
)
×
(
x
0
+
x
1
e
1
+
x
2
e
2
+
x
3
e
3
)
=
2
,
D
(
22
)
*
p
1
=
1
2
(
∂
∂
y
0
+
∂
∂
y
1
e
1
-
1
2
∂
∂
y
2
e
2
+
1
2
∂
∂
y
3
e
3
)
×
(
y
0
+
y
1
e
1
+
y
2
e
2
+
y
3
e
3
)
=
0
,
D
(
11
)
p
0
*
=
1
2
(
∂
∂
x
0
-
∂
∂
x
1
e
1
+
∂
∂
x
2
e
2
+
∂
∂
x
3
e
3
)
×
(
x
0
-
x
1
e
1
-
x
2
e
2
-
x
3
e
3
)
=
-
1
.
The other equations are calculated using a similar method, and the above equations are obtained.Theorem 7.
LetΩ be an open set in ℂ
2
×
ℂ
2 and let f
(
z
) be a function on Ω with values in 𝒟
(
𝒮
). Then the power z
n of z in 𝒟
(
𝒮
) is not an L
1-regular function but an L
2-regular function on Ω, where n
∈
ℕ.Proof.
From the definition of theL
r-regular function (
r
=
1,2
) on Ω and Proposition 6, we may consider whether the power z
n of z in 𝒟
(
𝒮
) satisfies the equation D
r
*
z
n
=
0
(
r
=
1,2
). Since D
(
11
)
*
p
0
=
2,
(44)
D
1
*
z
n
=
(
D
(
11
)
*
+
ε
D
(
12
)
*
)
(
p
0
n
+
ε
∑
k
=
1
n
‍
p
0
n
-
k
p
1
p
0
k
-
1
)
=
D
(
11
)
*
p
0
n
+
ε
(
∑
k
=
1
n
‍
D
(
11
)
*
p
0
n
-
k
p
1
p
0
k
-
1
+
D
(
12
)
*
p
0
n
)
≠
0
.
Hence, the power z
n of z is not L
1-regular on Ω. On the other hand, from the equations in Proposition 6, we have D
(
21
)
*
p
0
=
0, D
(
21
)
*
p
1
=
0, and D
(
22
)
*
p
0
=
0. Then,
(45)
D
2
*
z
n
=
D
(
21
)
*
p
0
n
+
ε
(
∑
k
=
1
n
‍
D
(
21
)
*
p
0
n
-
k
p
1
p
0
k
-
1
+
D
(
22
)
*
p
0
n
)
=
0
.
Therefore, by the definition of the L
r-regular function (
r
=
1,2
) on Ω, a power z
n of z is L
2-regular on Ω.



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*Source: 369430-2014-04-17.xml*