Research Article The Regularity of Functions on Dual Split …downloads.hindawi.com/journals/aaa/2014/369430.pdf · 2019-07-31 · Research Article The Regularity of Functions on

Research ArticleThe Regularity of Functions on Dual Split Quaternionsin Clifford Analysis

Ji Eun Kim and Kwang Ho Shon

Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea

Correspondence should be addressed to Kwang Ho Shon; [email protected]

Received 28 January 2014; Accepted 2 April 2014; Published 17 April 2014

Academic Editor: Zong-Xuan Chen

Copyright © 2014 J. E. Kim and K. H. Shon. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions andprovides differential operators in dual split quaternions and a dual split regular function on Ω ⊂ C2 × C2 that has a dual splitCauchy-Riemann system in dual split quaternions.

1. Introduction

Hamilton introduced quaternions, extending complex num-bers to higher spatial dimensions in differential geometry (see[1]). A set of quaternions can be represented as

H = {𝑧 = 𝑥0+ 𝑥1𝑖 + 𝑥2𝑗 + 𝑥3𝑘 : 𝑥𝑚

∈ R, 𝑚 = 0, 1, 2, 3} ,

(1)

where 𝑖2= 𝑗2= 𝑘2= −1, 𝑖𝑗𝑘 = −1, and R denotes the set of

real numbers. Cockle [2] introduced a set of split quaternionsas

S = {𝑧 = 𝑥0+ 𝑥1𝑒1+ 𝑥2𝑒2+ 𝑥3𝑒3: 𝑥𝑚

∈ R, 𝑚 = 0, 1, 2, 3} ,

(2)

where 𝑒2

1= −1, 𝑒

2

2= 𝑒2

3= 1, and 𝑒

1𝑒2𝑒3

= 1. Aset of split quaternions is noncommutative and containszero divisors, nilpotent elements, and nontrivial idempotents(see [3, 4]). Previous studies have examined the geometricand physical applications of split quaternions, which arerequired in solving split quaternionic equations (see [5, 6]).Inoguchi [7] reformulated the Gauss-Codazzi equations informs consistent with the theory of integrable systems in theMinkowski 3-space for split quaternion numbers.

A dual quaternion can be represented in a form reflect-ing an ordinary quaternion and a dual symbol. Because

dual-quaternion algebra is constructed from real eight-dimensional vector spaces and an ordered pair of quater-nions, dual quaternions are used in computer vision appli-cations. Kenwright [8] provided the characteristics of dualquaternions, and Pennestrı and Stefanelli [9] examined someproperties by using dual quaternions. Son [10, 11] offeredan extension problem for solutions of partial differentialequations and generalized solutions for the Riesz system. Byusing properties of Hamilton operators, Kula and Yayli [4]defined dual split quaternions and gave some properties ofthe screw motion in the Minkowski 3-space, showing thatHhas a rotation with unit split quaternions in H and a scalarproduct that allows it to be identifiedwith the semi-Euclideanspace for split quaternion numbers.

It was shown (see [12, 13]) that any complex-valued har-monic function 𝑓

1in a pseudoconvex domain 𝐷 of C2 × C2,

C being the set of complex numbers, has a conjugate function𝑓2in 𝐷 such that the quaternion-valued function 𝑓

1+ 𝑓2𝑗 is

hyperholomorphic in𝐷 and gave a regeneration theorem in aquaternion analysis in view of complex and Clifford analysis.In addition, we [14, 15] provided a new expression of thequaternionic basis and a regular function on reduced quater-nions by associating hypercomplex numbers 𝑒

1and 𝑒

2. We

[16] investigated the existence of hyperconjugate harmonicfunctions of an octonion number system, and we [17, 18]obtained some regular functions with values in dual quater-nions and researched an extension problem for properties

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 369430, 8 pageshttp://dx.doi.org/10.1155/2014/369430

2 Abstract and Applied Analysis

of regular functionswith values in dual quaternions and someapplications for such problems.

This paper provides a regular function and some prop-erties of differential operators in dual split quaternions. Inaddition,we research some equivalent conditions forCauchy-Riemann systems and expressions of power series in dual splitquaternions from the definition of dual split regular on anopen set Ω ⊂ C2 × C2.

2. Preliminaries

A dual number 𝐴 has the form 𝑎 + 𝜀𝑏, where 𝑎 and 𝑏 are realnumbers and 𝜀 is a dual symbol subject to the rules

𝜀 = 0, 0𝜀 = 𝜀0 = 0, 1𝜀 = 𝜀1 = 𝜀, 𝜀2= 0, (3)

and a split quaternion 𝑞 ∈ S is an expression of the form

𝑞 = 𝑥0+ 𝑥1𝑒1+ 𝑥2𝑒2+ 𝑥3𝑒3, (4)

where 𝑥𝑚

∈ R (𝑚 = 0, 1, 2, 3) and 𝑒𝑟(𝑟 = 1, 2, 3) are split

quaternionic units satisfying noncommutativemultiplicationrules (for split quaternions, see [1]):

𝑒2

1= −1, 𝑒

2

2= 𝑒2

3= 1,

𝑒1𝑒2= −𝑒2𝑒1= 𝑒3, 𝑒

2𝑒3= −𝑒3𝑒2= −𝑒1,

𝑒3𝑒1= −𝑒1𝑒3= 𝑒2.

(5)

Similarly, a dual split quaternion 𝑧 can be written as

D (S) = {𝑧 | 𝑧 = 𝑝0+ 𝜀𝑝1, 𝑝𝑟∈ S, 𝑟 = 0, 1} , (6)

which has elements of the following form:

𝑧 = {(𝑥0+ 𝑥1𝑒1) + (𝑥

2+ 𝑥3𝑒1) 𝑒2}

+ 𝜀 {(𝑦0+ 𝑦1𝑒1) + (𝑦

2+ 𝑦3𝑒1) 𝑒2}

= (𝑧0+ 𝑧1𝑒2) + 𝜀 (𝑧

2+ 𝑧3𝑒2)

= 𝑝0+ 𝜀𝑝1,

(7)

where 𝑝0= 𝑧0+ 𝑧1𝑒2and 𝑝

1= 𝑧2+ 𝑧3𝑒2are split quaternion

components, 𝑧0= 𝑥0+ 𝑥1𝑒1, 𝑧1= 𝑥2+ 𝑥3𝑒1, 𝑧2= 𝑦0+ 𝑦1𝑒1,

and 𝑧3= 𝑦2+ 𝑦3𝑒1are usual complex numbers, and 𝑥

𝑚, 𝑦𝑚

∈

R (𝑚 = 0, 1, 2, 3). The multiplication of split quaternionicunits with a dual symbol is commutative 𝜀𝑒

𝑟= 𝑒𝑟𝜀 (𝑟 =

1, 2, 3). However, by properties of split quaternionic unit,

𝑧𝑘𝑒𝑟= 𝑒𝑟𝑧𝑘

(𝑘 = 0, 1, 2, 3, 𝑟 = 0, 1) ,

𝑧𝑘𝑒𝑟= 𝑒𝑟𝑧𝑘

(𝑘 = 0, 1, 2, 3, 𝑟 = 2, 3) ,

𝑒𝑟𝑝𝑘

= 𝑝𝑘𝑒𝑟, 𝑒𝑟𝑝𝑘= 𝑝(𝑘𝑟)

𝑒𝑟

(𝑟 = 1, 2, 3, 𝑘 = 0, 1) ,

(8)

where

𝑝(01)

= 𝑧0− 𝑧1𝑒2= 𝑥0+ 𝑥1𝑒1− 𝑥2𝑒2− 𝑥3𝑒3,

𝑝(02)

= 𝑧0+ 𝑧1𝑒2= 𝑥0− 𝑥1𝑒1+ 𝑥2𝑒2− 𝑥3𝑒3,

𝑝(03)

= 𝑧0− 𝑧1𝑒2= 𝑥0− 𝑥1𝑒1− 𝑥2𝑒2+ 𝑥3𝑒3,

𝑝(11)

= 𝑧2− 𝑧3𝑒2= 𝑦0+ 𝑦1𝑒1− 𝑦2𝑒2− 𝑦3𝑒3,

𝑝(12)

= 𝑧2+ 𝑧3𝑒2= 𝑦0− 𝑦1𝑒1+ 𝑦2𝑒2− 𝑦3𝑒3,

𝑝(13)

= 𝑧2− 𝑧3𝑒2= 𝑦0− 𝑦1𝑒1− 𝑦2𝑒2+ 𝑦3𝑒3,

(9)

with 𝑧0

= 𝑥0− 𝑥1𝑒1, 𝑧1= 𝑥2− 𝑥3𝑒1, 𝑧2

= 𝑦0− 𝑦1𝑒1, and

𝑧3= 𝑦2− 𝑦3𝑒1. For instance,

𝑒2𝑝0= 𝑒2(𝑥0+ 𝑥1𝑒1+ 𝑥2𝑒2+ 𝑥3𝑒3)

= (𝑥0− 𝑥1𝑒1+ 𝑥2𝑒2− 𝑥3𝑒3) 𝑒2= 𝑝(02)

𝑒2,

𝑒1𝑝1= 𝑒1(𝑦0+ 𝑦1𝑒1+ 𝑦2𝑒2+ 𝑦3𝑒3)

= (𝑦0+ 𝑦1𝑒1− 𝑦2𝑒2− 𝑦3𝑒3) 𝑒1= 𝑝(11)

𝑒1.

(10)

Because of the properties of the eight-unit equality, the addi-tion and subtraction of dual split quaternions are governedby the rules of ordinary algebra. Here the symbol 𝑝

(𝑘𝑟)is used

by just enumerating 𝑟 and 𝑘, not 𝑟 times 𝑘. For example,𝑝(22)

= 𝑝4and 𝑝

22= 𝑝4.

For any two elements 𝑧 = 𝑝0+ 𝜀𝑝1and 𝑤 = 𝑞

0+ 𝜀𝑞1

of D(S), where 𝑞0= ∑3

𝑟=0𝑠𝑟𝑒𝑟and 𝑞

1= ∑3

𝑟=0𝑡𝑟𝑒𝑟are split

quaternion components and 𝑠𝑟, 𝑡𝑟∈ R (𝑟 = 0, 1, 2, 3), their

noncommutative product is given by

𝑧𝑤 = (𝑝0+ 𝜀𝑝1) (𝑞0+ 𝜀𝑞1) = 𝑝0𝑞0+ 𝜀 (𝑝

0𝑞1+ 𝑝1𝑞0) .

(11)

The conjugation 𝑧∗ of 𝑧 and the corresponding modulus 𝑧𝑧∗

inD(S) are defined by

𝑧∗= 𝑝∗

0+ 𝜀𝑝∗

1,

𝑧𝑧∗= 𝑧∗𝑧 = 𝑝0𝑝∗

0+ 𝜀 (𝑝

0𝑝∗

1+ 𝑝1𝑝∗

0)

= (𝑧0𝑧0− 𝑧1𝑧1) + 2𝜀 (𝑧

0𝑧2− 𝑧1𝑧3)

=

1

∑

𝑟=0

{(𝑥2

𝑟− 𝑥2

𝑟+2) + 𝜀 (𝑥

𝑟𝑦𝑟− 𝑥𝑟+2

𝑦𝑟+2

)} ,

(12)

where 𝑝∗0= 𝑧0− 𝑧1𝑒2and 𝑝

∗

1= 𝑧2− 𝑧3𝑒2.

Lemma 1. For all 𝑧 ∈ D(S) and 𝑛 ∈ N := {1, 2, 3, . . .}, wehave

𝑧𝑛= 𝑝𝑛

0+ 𝜀

𝑛

∑

𝑘=1

𝑝𝑛−𝑘

0𝑝1𝑝𝑘−1

0. (13)

Abstract and Applied Analysis 3

Proof. If 𝑛 = 1, then (13) is trivial. Now suppose that thisholds for some 𝑛 ∈ N. Then, as desired,

𝑧𝑛+1

= 𝑧𝑧𝑛= 𝑧(𝑝

𝑛

0+ 𝜀

𝑛

∑

𝑘=1

𝑝𝑛−𝑘

0𝑝1𝑝𝑘−1

0)

= 𝑝𝑛+1

0+ 𝜀

𝑛

∑

𝑘=1

𝑝𝑛−𝑘+1

0𝑝1𝑝𝑘−1

0+ 𝜀𝑝1𝑝𝑛

0

= 𝑝𝑛+1

0+ 𝜀

𝑛+1

∑

𝑘=1

𝑝𝑛+1−𝑘

0𝑝1𝑝𝑘−1

0.

(14)

By the principle of mathematical induction, (13) holds for all𝑛 ∈ N.

Let Ω be an open subset of C2 × C2. Then the function𝑓 : Ω → D(S) can be expressed as

𝑓 (𝑧) = 𝑓 (𝑝0, 𝑝1) = 𝑓0(𝑝0, 𝑝1) + 𝜀𝑓1(𝑝0, 𝑝1) , (15)

where the component functions 𝑓𝑟: Ω → S (𝑟 = 0, 1) are

split quaternionic-valued functions. The component func-tions 𝑓

𝑟(𝑟 = 0, 1) are

𝑓0(𝑝0, 𝑝1) = 𝑓0(𝑧0, 𝑧1, 𝑧2, 𝑧3)

= 𝑔0(𝑧0, 𝑧1, 𝑧2, 𝑧3) + 𝑔1(𝑧0, 𝑧1, 𝑧2, 𝑧3) 𝑒2,

𝑓1(𝑝0, 𝑝1) = 𝑓1(𝑧0, 𝑧1, 𝑧2, 𝑧3)

= 𝑔2(𝑧0, 𝑧1, 𝑧2, 𝑧3) + 𝑔3(𝑧0, 𝑧1, 𝑧2, 𝑧3) 𝑒2,

(16)

where 𝑔𝑘

= 𝑢2𝑘

+ 𝑢2𝑘+1

𝑒1(𝑘 = 0, 1) and 𝑔

𝑘= V2𝑘−4

+

V2𝑘−3

𝑒1(𝑘 = 2, 3) are complex-valued functions, and 𝑢

𝑟and

V𝑟(𝑟 = 0, 1, 2, 3) are real-valued functions.Now, we let differential operators 𝐷

1and 𝐷

2be defined

onD(S) as

𝐷1:= 𝐷(11)

+ 𝜀𝐷(12)

, 𝐷2:= 𝐷(21)

+ 𝜀𝐷(22)

. (17)

Then the conjugate operators𝐷∗1and𝐷

∗

2are

𝐷∗

1= 𝐷∗

(11)+ 𝜀𝐷∗

(12), 𝐷

∗

2= 𝐷∗

(21)+ 𝜀𝐷∗

(22), (18)

where

𝐷(11)

=𝜕

𝜕𝑧0

+𝜕

𝜕𝑧1

𝑒2=

1

2(

𝜕

𝜕𝑥0

−𝜕

𝜕𝑥1

𝑒1+

𝜕

𝜕𝑥2

𝑒2+

𝜕

𝜕𝑥3

𝑒3) ,

𝐷(12)

=𝜕

𝜕𝑧2

+𝜕

𝜕𝑧3

𝑒2=

1

2(

𝜕

𝜕𝑦0

−𝜕

𝜕𝑦1

𝑒1+

𝜕

𝜕𝑦2

𝑒2+

𝜕

𝜕𝑦3

𝑒3) ,

𝐷(21)

=𝜕

𝜕𝑧0

+1

2

𝜕

𝜕𝑧1

𝑒2

=1

2(

𝜕

𝜕𝑥0

−𝜕

𝜕𝑥1

𝑒1+

1

2

𝜕

𝜕𝑥2

𝑒2−

1

2

𝜕

𝜕𝑥3

𝑒3) ,

𝐷(22)

=𝜕

𝜕𝑧2

+1

2

𝜕

𝜕𝑧3

𝑒2

=1

2(

𝜕

𝜕𝑦0

−𝜕

𝜕𝑦1

𝑒1+

1

2

𝜕

𝜕𝑦2

𝑒2−

1

2

𝜕

𝜕𝑦3

𝑒3) ,

(19)

𝐷∗

(11)=

𝜕

𝜕𝑧0

−𝜕

𝜕𝑧1

𝑒2=

1

2(

𝜕

𝜕𝑥0

+𝜕

𝜕𝑥1

𝑒1−

𝜕

𝜕𝑥2

𝑒2−

𝜕

𝜕𝑥3

𝑒3) ,

𝐷∗

(12)=

𝜕

𝜕𝑧2

−𝜕

𝜕𝑧3

𝑒2=

1

2(

𝜕

𝜕𝑦0

+𝜕

𝜕𝑦1

𝑒1−

𝜕

𝜕𝑦2

𝑒2−

𝜕

𝜕𝑦3

𝑒3) ,

𝐷∗

(21)=

𝜕

𝜕𝑧0

−1

2

𝜕

𝜕𝑧1

𝑒2

=1

2(

𝜕

𝜕𝑥0

+𝜕

𝜕𝑥1

𝑒1−

1

2

𝜕

𝜕𝑥2

𝑒2+

1

2

𝜕

𝜕𝑥3

𝑒3) ,

𝐷∗

(22)=

𝜕

𝜕𝑧2

−1

2

𝜕

𝜕𝑧3

𝑒2

=1

2(

𝜕

𝜕𝑦0

+𝜕

𝜕𝑦1

𝑒1−

1

2

𝜕

𝜕𝑦2

𝑒2+

1

2

𝜕

𝜕𝑦3

𝑒3)

(20)

act on D(S). These operators are called correspondingCauchy-Riemann operators in D(S), where 𝜕/𝜕𝑧

𝑟and

𝜕/𝜕𝑧𝑟(𝑟 = 0, 1, 2, 3) are usual differential operators used in

the complex analysis.

Remark 2. From the definition of differential operators onD(S),

𝐷𝑟𝑓 = (𝐷

(𝑟1)+ 𝜀𝐷(𝑟2)

) (𝑓0+ 𝜀𝑓1)

= 𝐷(𝑟1)

𝑓0+ 𝜀 (𝐷

(𝑟1)𝑓1+ 𝐷(𝑟2)

𝑓0) ,

𝐷∗

𝑟𝑓 = (𝐷

∗

(𝑟1)+ 𝜀𝐷∗

(𝑟2)) (𝑓0+ 𝜀𝑓1)

= 𝐷∗

(𝑟1)𝑓0+ 𝜀 (𝐷

∗

(𝑟1)𝑓1+ 𝐷∗

(𝑟2)𝑓0) ,

(21)

where 𝑟 = 1, 2.

Definition 3. LetΩ be an open set inC2 ×C2. A function 𝑓 =

𝑓0+ 𝜀𝑓1is called an 𝐿

𝑟(resp., 𝑅

𝑟)-regular function (𝑟 = 1, 2)

onΩ if the following two conditions are satisfied:

(i) 𝑓𝑘(𝑘 = 0, 1) are continuously differential functions

onΩ, and

(ii) 𝐷∗𝑟𝑓(𝑧) = 0 (resp., 𝑓(𝑧)𝐷∗

𝑟= 0) onΩ (𝑟 = 1, 2).

In particular, the equation 𝐷∗

1𝑓(𝑧) = 0 of Definition 3 is

equivalent to

𝐷∗

(11)𝑓0= 0, 𝐷

∗

(12)𝑓0+ 𝐷∗

(11)𝑓1= 0. (22)


In addition,

𝜕𝑔0

𝜕𝑧0

−𝜕𝑔1

𝜕𝑧1

= 0,𝜕𝑔1

𝜕𝑧0

−𝜕𝑔0

𝜕𝑧1

= 0,

𝜕𝑔2

𝜕𝑧0

+𝜕𝑔0

𝜕𝑧2

−𝜕𝑔3

𝜕𝑧1

−𝜕𝑔1

𝜕𝑧3

= 0,

𝜕𝑔3

𝜕𝑧0

+𝜕𝑔1

𝜕𝑧2

−𝜕𝑔2

𝜕𝑧1

−𝜕𝑔0

𝜕𝑧3

= 0.

(23)

Concretely, the following system is obtained:

𝜕𝑢0

𝜕𝑥0

−𝜕𝑢1

𝜕𝑥1

−𝜕𝑢2

𝜕𝑥2

−𝜕𝑢3

𝜕𝑥3

= 0,

𝜕𝑢1

𝜕𝑥0

+𝜕𝑢0

𝜕𝑥1

−𝜕𝑢2

𝜕𝑥3

+𝜕𝑢3

𝜕𝑥2

= 0,

𝜕𝑢2

𝜕𝑥0

−𝜕𝑢3

𝜕𝑥1

−𝜕𝑢0

𝜕𝑥2

−𝜕𝑢1

𝜕𝑥3

= 0,

𝜕𝑢3

𝜕𝑥0

+𝜕𝑢2

𝜕𝑥1

−𝜕𝑢0

𝜕𝑥3

+𝜕𝑢1

𝜕𝑥2

= 0,

𝜕𝑢0

𝜕𝑦0

−𝜕𝑢1

𝜕𝑦1

−𝜕𝑢2

𝜕𝑦2

−𝜕𝑢3

𝜕𝑦3

+𝜕V0

𝜕𝑥0

−𝜕V1

𝜕𝑥1

−𝜕V2

𝜕𝑥2

−𝜕V3

𝜕𝑥3

= 0,

𝜕𝑢1

𝜕𝑦0

+𝜕𝑢0

𝜕𝑦1

−𝜕𝑢2

𝜕𝑦3

+𝜕𝑢3

𝜕𝑦2

+𝜕V1

𝜕𝑥0

+𝜕V0

𝜕𝑥1

−𝜕V2

𝜕𝑥3

+𝜕V3

𝜕𝑥2

= 0,

𝜕𝑢2

𝜕𝑦0

−𝜕𝑢3

𝜕𝑦1

−𝜕𝑢0

𝜕𝑦2

−𝜕𝑢1

𝜕𝑦3

+𝜕V2

𝜕𝑥0

−𝜕V3

𝜕𝑥1

−𝜕V0

𝜕𝑥2

−𝜕V1

𝜕𝑥3

= 0,

𝜕𝑢3

𝜕𝑦0

+𝜕𝑢2

𝜕𝑦1

−𝜕𝑢0

𝜕𝑦3

+𝜕𝑢1

𝜕𝑦2

+𝜕V3

𝜕𝑥0

+𝜕V2

𝜕𝑥1

−𝜕V0

𝜕𝑥3

+𝜕V1

𝜕𝑥2

= 0.

(24)

The above systems (23) and (24) are corresponding Cauchy-Riemann systems inD(S). Similarly, the equation𝐷

∗

2𝑓(𝑧) =

0 of Definition 3 is equivalent to

𝐷∗

(21)𝑓0= 0, 𝐷

∗

(22)𝑓0+ 𝐷∗

(21)𝑓1= 0. (25)

Then,

𝜕𝑔0

𝜕𝑧0

−1

2

𝜕𝑔1

𝜕𝑧1

= 0,𝜕𝑔1

𝜕𝑧0

−1

2

𝜕𝑔0

𝜕𝑧1

= 0,

𝜕𝑔2

𝜕𝑧0

+𝜕𝑔0

𝜕𝑧2

−1

2

𝜕𝑔3

𝜕𝑧1

−1

2

𝜕𝑔1

𝜕𝑧3

= 0,

𝜕𝑔3

𝜕𝑧0

+𝜕𝑔1

𝜕𝑧2

−1

2

𝜕𝑔2

𝜕𝑧1

−1

2

𝜕𝑔0

𝜕𝑧3

= 0.

(26)


𝜕𝑢0

𝜕𝑥0

−𝜕𝑢1

𝜕𝑥1

−1

2

𝜕𝑢2

𝜕𝑥2

+1

2

𝜕𝑢3

𝜕𝑥3

= 0,

𝜕𝑢1

𝜕𝑥0

+𝜕𝑢0

𝜕𝑥1

+1

2

𝜕𝑢2

𝜕𝑥3

+1

2

𝜕𝑢3

𝜕𝑥2

= 0,

𝜕𝑢2

𝜕𝑥0

−𝜕𝑢3

𝜕𝑥1

−1

2

𝜕𝑢0

𝜕𝑥2

+1

2

𝜕𝑢1

𝜕𝑥3

= 0,

𝜕𝑢3

𝜕𝑥0

+𝜕𝑢2

𝜕𝑥1

+1

2

𝜕𝑢0

𝜕𝑥3

+1

2

𝜕𝑢1

𝜕𝑥2

= 0,

𝜕𝑢0

𝜕𝑦0

−𝜕𝑢1

𝜕𝑦1

−1

2

𝜕𝑢2

𝜕𝑦2

+1

2

𝜕𝑢3

𝜕𝑦3

+𝜕V0

𝜕𝑥0

−𝜕V1

𝜕𝑥1

−1

2

𝜕V2

𝜕𝑥2

+1

2

𝜕V3

𝜕𝑥3

= 0,

𝜕𝑢1

𝜕𝑦0

+𝜕𝑢0

𝜕𝑦1

+1

2

𝜕𝑢2

𝜕𝑦3

+1

2

𝜕𝑢3

𝜕𝑦2

+𝜕V1

𝜕𝑥0

+𝜕V0

𝜕𝑥1

+1

2

𝜕V2

𝜕𝑥3

+1

2

𝜕V3

𝜕𝑥2

= 0,

𝜕𝑢2

𝜕𝑦0

−𝜕𝑢3

𝜕𝑦1

−1

2

𝜕𝑢0

𝜕𝑦2

+1

2

𝜕𝑢1

𝜕𝑦3

+𝜕V2

𝜕𝑥0

−𝜕V3

𝜕𝑥1

−1

2

𝜕V0

𝜕𝑥2

+1

2

𝜕V1

𝜕𝑥3

= 0,

𝜕𝑢3

𝜕𝑦0

+𝜕𝑢2

𝜕𝑦1

+1

2

𝜕𝑢0

𝜕𝑦3

+1

2

𝜕𝑢1

𝜕𝑦2

+𝜕V3

𝜕𝑥0

+𝜕V2

𝜕𝑥1

+1

2

𝜕V0

𝜕𝑥3

+1

2

𝜕V1

𝜕𝑥2

= 0.

(27)

The above systems (26) and (27) are corresponding Cauchy-Riemann systems inD(S).

On the other hand, the equation 𝑓(𝑧)𝐷∗

1= 0 of

Definition 3 is equivalent to

𝑓0𝐷∗

(11)= 0, 𝑓

0𝐷∗

(12)= −𝑓1𝐷∗

(11). (28)


Then,

𝑔0

𝜕

𝜕𝑧0

= 𝑔1

𝜕

𝜕𝑧1

, 𝑔1

𝜕

𝜕𝑧0

= 𝑔0

𝜕

𝜕𝑧1

,

𝑔0

𝜕

𝜕𝑧2

− 𝑔1

𝜕

𝜕𝑧3

= − 𝑔2

𝜕

𝜕𝑧0

+ 𝑔3

𝜕

𝜕𝑧1

,

𝑔1

𝜕

𝜕𝑧2

− 𝑔0

𝜕

𝜕𝑧3

= − 𝑔3

𝜕

𝜕𝑧0

+ 𝑔2

𝜕

𝜕𝑧1

,

(29)

where

𝑔𝑘

𝜕

𝜕𝑧𝑚

=𝜕𝑔𝑘

𝜕𝑧𝑚

, 𝑔𝑘

𝜕

𝜕𝑧𝑚

=𝜕𝑔𝑘

𝜕𝑧𝑚

(𝑘,𝑚 = 0, 1, 2, 3) .

(30)


𝜕𝑢0

𝜕𝑥0

−𝜕𝑢1

𝜕𝑥1

=𝜕𝑢2

𝜕𝑥2

+𝜕𝑢3

𝜕𝑥3

,𝜕𝑢1

𝜕𝑥0

+𝜕𝑢0

𝜕𝑥1

= −𝜕𝑢2

𝜕𝑥3

+𝜕𝑢3

𝜕𝑥2

,

𝜕𝑢2

𝜕𝑥0

+𝜕𝑢3

𝜕𝑥1

=𝜕𝑢0

𝜕𝑥2

−𝜕𝑢1

𝜕𝑥3

,𝜕𝑢3

𝜕𝑥0

−𝜕𝑢2

𝜕𝑥1

=𝜕𝑢0

𝜕𝑥3

+𝜕𝑢1

𝜕𝑥2

,

𝜕𝑢0

𝜕𝑦0

−𝜕𝑢1

𝜕𝑦1

−𝜕𝑢2

𝜕𝑦2

−𝜕𝑢3

𝜕𝑦3

= −𝜕V0

𝜕𝑥0

+𝜕V1

𝜕𝑥1

+𝜕V2

𝜕𝑥2

+𝜕V3

𝜕𝑥3

,

𝜕𝑢1

𝜕𝑦0

+𝜕𝑢0

𝜕𝑦1

+𝜕𝑢2

𝜕𝑦3

−𝜕𝑢3

𝜕𝑦2

= −𝜕V1

𝜕𝑥0

−𝜕V0

𝜕𝑥1

−𝜕V2

𝜕𝑥3

+𝜕V3

𝜕𝑥2

,

𝜕𝑢2

𝜕𝑦0

+𝜕𝑢3

𝜕𝑦1

−𝜕𝑢0

𝜕𝑦2

+𝜕𝑢1

𝜕𝑦3

= −𝜕V2

𝜕𝑥0

−𝜕V3

𝜕𝑥1

+𝜕V0

𝜕𝑥2

−𝜕V1

𝜕𝑥3

,

𝜕𝑢3

𝜕𝑦0

−𝜕𝑢2

𝜕𝑦1

−𝜕𝑢0

𝜕𝑦3

−𝜕𝑢1

𝜕𝑦2

= −𝜕V3

𝜕𝑥0

+𝜕V2

𝜕𝑥1

+𝜕V0

𝜕𝑥3

+𝜕V1

𝜕𝑥2

.

(31)

Similarly, the equation 𝑓(𝑧)𝐷∗

2= 0 of Definition 3 is

equivalent to

𝑓0𝐷∗

(21)= 0, 𝑓

0𝐷∗

(22)= −𝑓1𝐷∗

(21). (32)

Then,

𝑔0

𝜕

𝜕𝑧0

=1

2𝑔1

𝜕

𝜕𝑧1

, 𝑔1

𝜕

𝜕𝑧0

=1

2𝑔0

𝜕

𝜕𝑧1

,

𝑔0

𝜕

𝜕𝑧2

−1

2𝑔1

𝜕

𝜕𝑧3

= − 𝑔2

𝜕

𝜕𝑧0

+1

2𝑔3

𝜕

𝜕𝑧1

,

𝑔1

𝜕

𝜕𝑧2

−1

2𝑔0

𝜕

𝜕𝑧3

= − 𝑔3

𝜕

𝜕𝑧0

+1

2𝑔2

𝜕

𝜕𝑧1

,

(33)

where

𝑔𝑘

𝜕

𝜕𝑧𝑚

=𝜕𝑔𝑘

𝜕𝑧𝑚

, 𝑔𝑘

𝜕

𝜕𝑧𝑚

=𝜕𝑔𝑘

𝜕𝑧𝑚

(𝑘,𝑚 = 0, 1, 2, 3) .

(34)

Concretely, the system is obtained as follows:

𝜕𝑢0

𝜕𝑥0

−𝜕𝑢1

𝜕𝑥1

−1

2

𝜕𝑢2

𝜕𝑥2

+1

2

𝜕𝑢3

𝜕𝑥3

= 0,

𝜕𝑢1

𝜕𝑥0

+𝜕𝑢0

𝜕𝑥1

−1

2

𝜕𝑢2

𝜕𝑥3

−1

2

𝜕𝑢3

𝜕𝑥2

= 0,

𝜕𝑢2

𝜕𝑥0

+𝜕𝑢3

𝜕𝑥1

−1

2

𝜕𝑢0

𝜕𝑥2

−1

2

𝜕𝑢1

𝜕𝑥3

= 0,

𝜕𝑢3

𝜕𝑥0

−𝜕𝑢2

𝜕𝑥1

+1

2

𝜕𝑢0

𝜕𝑥3

−1

2

𝜕𝑢1

𝜕𝑥2

= 0,

𝜕𝑢0

𝜕𝑦0

−𝜕𝑢1

𝜕𝑦1

−1

2

𝜕𝑢2

𝜕𝑦2

+1

2

𝜕𝑢3

𝜕𝑦3

+𝜕V0

𝜕𝑥0

−𝜕V1

𝜕𝑥1

−1

2

𝜕V2

𝜕𝑥2

+1

2

𝜕V3

𝜕𝑥3

= 0,

𝜕𝑢1

𝜕𝑦0

+𝜕𝑢0

𝜕𝑦1

−1

2

𝜕𝑢2

𝜕𝑦3

−1

2

𝜕𝑢3

𝜕𝑦2

+𝜕V1

𝜕𝑥0

+𝜕V0

𝜕𝑥1

−1

2

𝜕V2

𝜕𝑥3

−1

2

𝜕V3

𝜕𝑥2

= 0,

𝜕𝑢2

𝜕𝑦0

+𝜕𝑢3

𝜕𝑦1

−1

2

𝜕𝑢0

𝜕𝑦2

+1

2

𝜕𝑢1

𝜕𝑦3

+𝜕V2

𝜕𝑥0

+𝜕V3

𝜕𝑥1

−1

2

𝜕V0

𝜕𝑥2

+1

2

𝜕V1

𝜕𝑥3

= 0,

𝜕𝑢3

𝜕𝑦0

−𝜕𝑢2

𝜕𝑦1

−1

2

𝜕𝑢0

𝜕𝑦3

−1

2

𝜕𝑢1

𝜕𝑦2

+𝜕V3

𝜕𝑥0

−𝜕V2

𝜕𝑥1

−1

2

𝜕V0

𝜕𝑥3

−1

2

𝜕V1

𝜕𝑥2

= 0.

(35)

From the systems (24), (27), (31), and (35), the equations𝐷∗

𝑟𝑓(𝑧) = 0 and 𝑓(𝑧)𝐷

∗

𝑟= 0 (𝑟 = 1, 2) are different.

Therefore, the equations 𝐷∗𝑟𝑓(𝑧) = 0 and 𝑓(𝑧)𝐷

∗

𝑟= 0 (𝑟 =

1, 2) should be distinguished as 𝐿𝑟-regular functions (𝑟 =

1, 2) and 𝑅𝑟-regular functions (𝑟 = 1, 2) on Ω, respectively.

Now the properties of the 𝐿𝑟-regular function (𝑟 = 1, 2) with

values inD(S) are considered.

3. Properties of 𝐿𝑟-Regular Functions (𝑟 = 1, 2)

with Values in D(S)

We consider properties of a 𝐿𝑟-regular functions (𝑟 = 1, 2)

with values inD(S).

Theorem 4. Let Ω be an open set in C2 × C2 and let 𝑓 =

𝑓0+ 𝜀𝑓1= (𝑔0+𝑔1𝑒2)+ 𝜀(𝑔

2+𝑔3𝑒2) be an 𝐿

1-regular function

defined on Ω. Then

𝐷1𝑓 = {2(

𝜕

𝜕𝑧1

+ 𝜀𝜕

𝜕𝑧3

) 𝑒2− (

𝜕

𝜕𝑥1

+ 𝜀𝜕

𝜕𝑦1

) 𝑒1}𝑓. (36)


Proof. By the system (23), we have

𝐷1𝑓 = 𝐷

(11)𝑓0+ 𝜀 (𝐷

(12)𝑓0+ 𝐷(11)

𝑓1)

= (𝜕𝑔0

𝜕𝑧0

+𝜕𝑔1

𝜕𝑧1

) + (𝜕𝑔1

𝜕𝑧0

+𝜕𝑔0

𝜕𝑧1

) 𝑒2

+ 𝜀(𝜕𝑔0

𝜕𝑧2

+𝜕𝑔1

𝜕𝑧3

+𝜕𝑔2

𝜕𝑧0

+𝜕𝑔3

𝜕𝑧1

)

+ 𝜀(𝜕𝑔1

𝜕𝑧2

+𝜕𝑔0

𝜕𝑧3

+𝜕𝑔3

𝜕𝑧0

+𝜕𝑔2

𝜕𝑧1

) 𝑒2

= (𝜕𝑔0

𝜕𝑧0

+𝜕𝑢1

𝜕𝑥1

−𝜕𝑢0

𝜕𝑥1

𝑒1+

𝜕𝑔1

𝜕𝑧1

)

+ (𝜕𝑔1

𝜕𝑧0

+𝜕𝑢3

𝜕𝑥1

−𝜕𝑢2

𝜕𝑥1

𝑒1+

𝜕𝑔0

𝜕𝑧1

) 𝑒2

+ 𝜀(𝜕𝑔0

𝜕𝑧2

+𝜕𝑢1

𝜕𝑦1

−𝜕𝑢0

𝜕𝑦1

𝑒1+

𝜕𝑔1

𝜕𝑧3

+𝜕𝑔2

𝜕𝑧0

+𝜕V1

𝜕𝑥1

−𝜕V0

𝜕𝑥1

𝑒1+

𝜕𝑔3

𝜕𝑧1

)

+ 𝜀(𝜕𝑔1

𝜕𝑧2

+𝜕𝑢3

𝜕𝑦1

−𝜕𝑢2

𝜕𝑦1

𝑒1+

𝜕𝑔0

𝜕𝑧3

+𝜕𝑔3

𝜕𝑧0

+𝜕V3

𝜕𝑥1

−𝜕V2

𝜕𝑥1

𝑒1+

𝜕𝑔2

𝜕𝑧1

) 𝑒2

= (𝜕𝑢1

𝜕𝑥1

−𝜕𝑢0

𝜕𝑥1

𝑒1+ 2

𝜕𝑔1

𝜕𝑧1

)

+ (𝜕𝑢3

𝜕𝑥1

−𝜕𝑢2

𝜕𝑥1

𝑒1+ 2

𝜕𝑔0

𝜕𝑧1

) 𝑒2

+ 𝜀(𝜕𝑢1

𝜕𝑦1

−𝜕𝑢0

𝜕𝑦1

𝑒1+ 2

𝜕𝑔1

𝜕𝑧3

+𝜕V1

𝜕𝑥1

−𝜕V0

𝜕𝑥1

𝑒1+ 2

𝜕𝑔3

𝜕𝑧1

)

+ 𝜀(𝜕𝑢3

𝜕𝑦1

−𝜕𝑢2

𝜕𝑦1

𝑒1+ 2

𝜕𝑔0

𝜕𝑧3

+𝜕V3

𝜕𝑥1

−𝜕V2

𝜕𝑥1

𝑒1+ 2

𝜕𝑔2

𝜕𝑧1

) 𝑒2

= {2(𝜕

𝜕𝑧1

+ 𝜀𝜕

𝜕𝑧3

) 𝑒2− (

𝜕

𝜕𝑥1

+ 𝜀𝜕

𝜕𝑦1

) 𝑒1}𝑓.

(37)

Therefore, we obtain

𝐷1𝑓 = {2(

𝜕

𝜕𝑧1

+ 𝜀𝜕

𝜕𝑧3

) 𝑒2− (

𝜕

𝜕𝑥1

+ 𝜀𝜕

𝜕𝑦1

) 𝑒1}𝑓. (38)

Theorem5. LetΩ be an open set inC2×C2 and𝑓 = 𝑓0+𝜀𝑓1=

(𝑔0+𝑔1𝑒2) + 𝜀(𝑔

2+𝑔3𝑒2) be an 𝐿

2-regular function defined on

Ω. Then

𝐷2𝑓 = {(

𝜕

𝜕𝑧1

+ 𝜀𝜕

𝜕𝑧3

) 𝑒2− (

𝜕

𝜕𝑥1

+ 𝜀𝜕

𝜕𝑦1

) 𝑒1}𝑓. (39)

Proof. By the system (26), we have

𝐷2𝑓 = 𝐷

(21)𝑓0+ 𝜀 (𝐷

(22)𝑓0+ 𝐷(21)

𝑓1)

= (𝜕𝑔0

𝜕𝑧0

+1

2

𝜕𝑔1

𝜕𝑧1

) + (𝜕𝑔1

𝜕𝑧0

+1

2

𝜕𝑔0

𝜕𝑧1

) 𝑒2

+ 𝜀(𝜕𝑔0

𝜕𝑧2

+1

2

𝜕𝑔1

𝜕𝑧3

+𝜕𝑔2

𝜕𝑧0

+1

2

𝜕𝑔3

𝜕𝑧1

)

+ 𝜀(𝜕𝑔1

𝜕𝑧2

+1

2

𝜕𝑔0

𝜕𝑧3

+𝜕𝑔3

𝜕𝑧0

+1

2

𝜕𝑔2

𝜕𝑧1

) 𝑒2

= (𝜕𝑔0

𝜕𝑧0

+𝜕𝑢1

𝜕𝑥1

−𝜕𝑢0

𝜕𝑥1

𝑒1+

1

2

𝜕𝑔1

𝜕𝑧1

)

+ (𝜕𝑔1

𝜕𝑧0

+𝜕𝑢3

𝜕𝑥1

−𝜕𝑢2

𝜕𝑥1

𝑒1+

1

2

𝜕𝑔0

𝜕𝑧1

) 𝑒2

+ 𝜀(𝜕𝑔0

𝜕𝑧2

+𝜕𝑢1

𝜕𝑦1

−𝜕𝑢0

𝜕𝑦1

𝑒1+

1

2

𝜕𝑔1

𝜕𝑧3

+𝜕𝑔2

𝜕𝑧0

+𝜕V1

𝜕𝑥1

−𝜕V0

𝜕𝑥1

𝑒1+

1

2

𝜕𝑔3

𝜕𝑧1

)

+ 𝜀(𝜕𝑔1

𝜕𝑧2

+𝜕𝑢3

𝜕𝑦1

−𝜕𝑢2

𝜕𝑦1

𝑒1+

1

2

𝜕𝑔0

𝜕𝑧3

+𝜕𝑔3

𝜕𝑧0

+𝜕V3

𝜕𝑥1

−𝜕V2

𝜕𝑥1

𝑒1+

1

2

𝜕𝑔2

𝜕𝑧1

) 𝑒2

= (𝜕𝑢1

𝜕𝑥1

−𝜕𝑢0

𝜕𝑥1

𝑒1+

𝜕𝑔1

𝜕𝑧1

)

+ (𝜕𝑢3

𝜕𝑥1

−𝜕𝑢2

𝜕𝑥1

𝑒1+

𝜕𝑔0

𝜕𝑧1

) 𝑒2

+ 𝜀(𝜕𝑢1

𝜕𝑦1

−𝜕𝑢0

𝜕𝑦1

𝑒1+

𝜕𝑔1

𝜕𝑧3

+𝜕V1

𝜕𝑥1

−𝜕V0

𝜕𝑥1

𝑒1+

𝜕𝑔3

𝜕𝑧1

)

+ 𝜀(𝜕𝑢3

𝜕𝑦1

−𝜕𝑢2

𝜕𝑦1

𝑒1+

𝜕𝑔0

𝜕𝑧3

+𝜕V3

𝜕𝑥1

−𝜕V2

𝜕𝑥1

𝑒1+

𝜕𝑔2

𝜕𝑧1

) 𝑒2

= {(𝜕

𝜕𝑧1

+ 𝜀𝜕

𝜕𝑧3

) 𝑒2− (

𝜕

𝜕𝑥1

+ 𝜀𝜕

𝜕𝑦1

) 𝑒1}𝑓.

(40)

Therefore, we obtain the following equation:

𝐷2𝑓 = {(

𝜕

𝜕𝑧1

+ 𝜀𝜕

𝜕𝑧3

) 𝑒2− (

𝜕

𝜕𝑥1

+ 𝜀𝜕

𝜕𝑦1

) 𝑒1}𝑓. (41)


Proposition 6. From properties of differential operators, thefollowing equations are obtained:

𝐷(1𝑟)

𝑝𝑟−1

= 2, 𝐷(2𝑟)

𝑝𝑟−1

= 1,

𝐷∗

(1𝑟)𝑝𝑟−1

= −1, 𝐷∗

(2𝑟)𝑝𝑟−1

= 0,

𝐷∗

(1𝑟)𝑝∗

𝑟−1= 2, 𝐷

∗

(2𝑟)𝑝∗

𝑟−1= 1,

𝐷(𝑟1)

𝑝1= 𝐷∗

(𝑟1)𝑝1= 𝐷(𝑟1)

𝑝∗

1= 𝐷∗

(𝑟1)𝑝∗

1

= 𝐷(𝑟2)

𝑝0= 𝐷∗

(𝑟2)𝑝0= 𝐷(𝑟2)

𝑝∗

0

= 𝐷∗

(𝑟2)𝑝∗

0= 0 (𝑟 = 1, 2) .

(42)

Proof. By properties of the power of dual split quaternionsand derivatives on D(S), the following derivatives areobtained:

𝐷(11)

𝑝0=

1

2(

𝜕

𝜕𝑥0

−𝜕

𝜕𝑥1

𝑒1+

𝜕

𝜕𝑥2

𝑒2+

𝜕

𝜕𝑥3

𝑒3)

× (𝑥0+ 𝑥1𝑒1+ 𝑥2𝑒2+ 𝑥3𝑒3) = 2,

𝐷∗

(22)𝑝1=

1

2(

𝜕

𝜕𝑦0

+𝜕

𝜕𝑦1

𝑒1−

1

2

𝜕

𝜕𝑦2

𝑒2+

1

2

𝜕

𝜕𝑦3

𝑒3)

× (𝑦0+ 𝑦1𝑒1+ 𝑦2𝑒2+ 𝑦3𝑒3) = 0,

𝐷(11)

𝑝∗

0=

1

2(

𝜕

𝜕𝑥0

−𝜕

𝜕𝑥1

𝑒1+

𝜕

𝜕𝑥2

𝑒2+

𝜕

𝜕𝑥3

𝑒3)

× (𝑥0− 𝑥1𝑒1− 𝑥2𝑒2− 𝑥3𝑒3) = −1.

(43)

The other equations are calculated using a similar method,and the above equations are obtained.

Theorem 7. LetΩ be an open set in C2 ×C2 and let 𝑓(𝑧) be afunction on Ω with values inD(S). Then the power 𝑧𝑛 of 𝑧 inD(S) is not an 𝐿

1-regular function but an 𝐿

2-regular function

on Ω, where 𝑛 ∈ N.

Proof. From the definition of the 𝐿𝑟-regular function (𝑟 =

1, 2) on Ω and Proposition 6, we may consider whether thepower 𝑧𝑛 of 𝑧 in D(S) satisfies the equation 𝐷

∗

𝑟𝑧𝑛= 0 (𝑟 =

1, 2). Since𝐷∗(11)

𝑝0= 2,

𝐷∗

1𝑧𝑛= (𝐷∗

(11)+ 𝜀𝐷∗

(12))(𝑝𝑛

0+ 𝜀

𝑛

∑

𝑘=1

𝑝𝑛−𝑘

0𝑝1𝑝𝑘−1

0)

= 𝐷∗

(11)𝑝𝑛

0+ 𝜀(

𝑛

∑

𝑘=1

𝐷∗

(11)𝑝𝑛−𝑘

0𝑝1𝑝𝑘−1

0+ 𝐷∗

(12)𝑝𝑛

0) = 0.

(44)

Hence, the power 𝑧𝑛 of 𝑧 is not 𝐿1-regular onΩ. On the other

hand, from the equations in Proposition 6, we have𝐷∗(21)

𝑝0=

0,𝐷∗(21)

𝑝1= 0, and𝐷

∗

(22)𝑝0= 0. Then,

𝐷∗

2𝑧𝑛= 𝐷∗

(21)𝑝𝑛

0+ 𝜀(

𝑛

∑

𝑘=1

𝐷∗

(21)𝑝𝑛−𝑘

0𝑝1𝑝𝑘−1

0+ 𝐷∗

(22)𝑝𝑛

0) = 0.

(45)

Therefore, by the definition of the 𝐿𝑟-regular function (𝑟 =

1, 2) onΩ, a power 𝑧𝑛 of 𝑧 is 𝐿2-regular onΩ.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The second author was supported by the Basic ScienceResearch Program through the National Research Founda-tion of Korea (NRF) funded by the Ministry of Science, ICT,and Future Planning (2013R1A1A2008978).

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