CHARACTERIZATION OF A REGULAR FUNCTION WITH VALUES IN DUAL QUATERNIONS

The Pure and Applied Mathematics.
2015.
Feb,
22(1):
65-74

- Received : October 31, 2014
- Accepted : January 10, 2015
- Published : February 28, 2015

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In this paper, we provide the notions of dual quaternions and their algebraic properties based on matrices. From quaternion analysis, we give the concept of a derivative of functions and and obtain a dual quaternion Cauchy-Riemann system that are equivalent. Also, we research properties of a regular function with values in dual quaternions and relations derivative with a regular function in dual quaternions.
T
be the set of quaternion numbers constructed over a real Euclidean quadratic four dimensional vector space. In 2004 and 2006, Kajiwara, Li and Shon
[2
,
3]
obtained some results for the regeneration in complex, quaternion and Clifford analysis, and for the inhomogeneous Cauchy-Riemann system of quaternions and Clifford analysis in ellipsoid. Naser
[12]
and Nôno
[13]
obtained some properties of quaternionic hyperholomorphic functions. In 2011, Koriyama, Mae and Nôno
[8
,
9]
researched for hyperholomorphic functions and holomorphic functions in quaternion analysis. Also, they obtained some results of regularities of octonion functions and holomorphic mappings. In 2012, Gotô and Nôno
[1]
researched for regular functions with values in a commutative subalgebra
of matrix algebra
. Lim and Shon
[10
,
11]
obtained some properties of hyperholomorphic functions and researched for the hyperholomophic functions and hyperconjugate harmonic functions of octonion variables, and for the dual quaternion functions and its applications. Recently, we
[4
,
5
,
6
,
7]
obtained some results for the regularity of functions on the ternary quaternion and reduced quaternion field in Clifford analysis, and for the regularity of functions on dual split quaternions in Clifford analysis. Also, we investigated the corresponding Cauchy-Riemann systems in special quaternions and properties of each regular functions defined by the corresponding differential operators in special quaternions.
The aim of the paper is to define the representations of dual quaternions, written by a matrix form. Also, we research the conditions of the derivative of functions with values in dual quaternions and the definition of a regular function for Cauchy-Riemann system in dual quaternions.
T
of quaternions
is a four dimensional non-commutative real field such that its four base elements
e
_{0}
= 1,
e
_{1}
,
e
_{2}
and
e
_{3}
satisfying the following :
The element
e
_{0}
= 1 is the identity of
T
. Identifying the element
e
_{1}
with the imaginary unit
in the complex field of complex numbers. The dual numbers extended the real numbers by adjoining one new non-zero element
ε
with the property
ε
^{2}
= 0. The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form
z
=
x
+
εy
with
x
and
y
uniquely determined real numbers. Dual numbers form the coefficients of dual quaternions. If we use matrices, dual numbers can be represented as
The sum and product of dual numbers are then calculated with ordinary matrix addition and matrix multiplication; both operations are commutative and associative within the algebra of dual numbers.
where
, is a non-commutative subalgebra of
.
We define that the dual quaternionic multiplication of two dual quaternions
and
is given by
The dual quaternionic conjugate
Z
* of
Z
is
Then the modulus |
Z
| and the inverse
Z
^{−1}
of Z in
are defined by the following :
and
By using the multiplication of
, the power of
Z
is for
,
and the division of two
can be computed as
Since
and
are real variables, it can be written by
where
and
We use the following differential operators :
where
(
k
= 1, 2) are usual complex differential operations.
The Laplacian operator is
Let
S
be a bounded open subset in
T
×
T
. A function
F
(
Z
) is defined by the following form in
S
with values in
:
where
u_{j}
=
u_{j}
(
x
_{0}
,
x
_{1}
,
x
_{2}
,
x
_{3}
,
y
_{0}
,
y
_{1}
,
y
_{2}
,
y
_{3}
) and
v_{j}
=
v_{j}
(
x
_{0}
,
x
_{1}
,
x
_{2}
,
x
_{3}
,
y
_{0}
,
y
_{1}
,
y
_{2}
,
y
_{3}
) are real valued functions.
Remark 3.1.
Using differential operators, we have the following equations:
where
Definition 3.2.
Let
S
be a bounded open subset in
T
×
T
. A function
F
=
f
+
εg
is said to be M-
regular
in
S
if
f
and
g
of
F
are continuously differential quaternion valued functions in
S
such that
D
*
F
= 0.
Remark 3.3.
The equation
D
*
F
= 0 is equivalent to
Also, it is equivalent to
The above system is called a dual quaternion Cauchy-Riemann system in dual quaternions.
Let Ω be an open subset of
, for
Z
_{0}
=
z
_{0}
+
εω
_{0}
∈ Ω,
is called a dual-quaternion function in
.
Definition 3.4.
A function
F
is said to be
continuous
at
Z
_{0}
=
z
_{0}
+
εω
_{0}
if
where the limit has
Definition 3.5.
The dual quaternion function
F
is said to be
differentiable
in dual quaternions if the limit
exists and the limit is called the derivative of
F
in dual quaternions.
Remark 3.6.
From the definition of derivative of
f
and properties of differential operations of quaternion valued functions, we have
where
is a constant in a domain of
f
(see
[2
,
11]
). Since the equation (3.2) is equivalent to
D_{z}f
, we can express
. Hence, by the representations of
DF
and properties of limit, calculating the division for
Therefore, we can represent
Theorem 3.7.
Let F = f + εg be a dual quaternion function in
.
If F satisfies the equation Df
= 0,
then the derivative of F satisfies the following equation:
Proof.
By the division of dual quaternions, we have
Then, the limit
exists if and only if
has two cases to deal with
Case 1)
If
then the limit exists and the derivative can be written by
Case 2)
If
then the limit exists and the derivative can be written by
Therefore, the equation
is obtained.
Theorem 3.8.
Let F
=
f
+
εg be a dual quaternion function in
.
If F is a M-regular function in dual quaternions, that is, F satisfies the equation D
*
F
= 0,
then the derivative of F satisfies the following equation:
Proof.
From the proof of Theorem 3.7, we have
Since
F
satisfies a dual quaternion Cauchy-Riemann system (3.1), we have
Therefore, since
we have

1. INTRODUCTION

Let
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2. PRELIMINARIES

The field
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3. DUAL QUATERNIONS

The algebra
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Gotô S.
,
Nôno K.
2012
Regular Functions with Values in a Commutative Subalgebra ℂ(ℂ) of Matrix Algebra M(4;ℝ)
Bull. Fukuoka Univ. Ed. Part 3
61
9 -
15

Kajiwara J.
,
Li X.D.
,
Shon K.H.
2004
Regeneration in complex, quaternion and Clifford analysis
Kluwer Acad. Pub.
Proc. the 9th International Conf. on Finite or Infinite Dimensional Complex Analysis and Applications, Adv. Complex Anal. Appl.
Hanoi
(9)
287 -
298

Kajiwara J.
,
Li X.D.
,
Shon K.H.
2006
Function spaces in complex and Clifford analysis, Inhomogeneous Cauchy Riemann system of quaternion and Clifford analysis in ellipsoid
National Univ. Pub.
Proc. the 14th International Conf. on Finite or Infinite Dimensional Complex Analysis and Applications
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14
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155

Kim J.E.
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Lim S.J.
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Shon K.H.
2013
Taylor series of functions with values in dual quaternion
J. Korean Soc. Math. Educ., Ser. B, Pure Appl. Math.
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251 -
258

Kim J.E.
,
Lim S.J.
,
Shon K.H.
2013
Regular functions with values in ternary number system on the complex Clifford analysis
Abstr. Appl. Anal.
Article ID. 136120
2013
7 -

Kim J.E.
,
Lim S.J.
,
Shon K.H.
2014
Regularity of functions on the reduced quaternion field in Clifford analysis
Abstr. Appl. Anal.
Article ID. 654798
2014
8 -

Kim J.E.
,
Lim S.J.
,
Shon K.H.
2014
The Regularity of functions on Dual split quaternions in Clifford analysis
Abstr. Appl. Anal.
Article ID. 369430
2014
8 -

Koriyama H.
,
Mae H.
,
Nôno K.
2011
Hyperholomorphic fucntions and holomorphic functions in quaternionic analysis
Bull. Fukuoka Univ. Ed. part III
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9

Koriyama H.
,
Mae H.
,
Nôno K.
2011
On regularities of octonionic functions and holomorphic mappings
Bull. Fukuoka Univ. Ed. part III
60
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28

Lim S.J.
,
Shon K.H.
2012
Properties of hyperholomorphic functions in Clifford analysis
East Asian Math. J.
28
(5)
553 -
559
** DOI : 10.7858/eamj.2012.040**

Lim S.J.
,
Shon K.H.
2013
Dual quaternion functions and its applications
J. Applied Math.
Article ID 583813
6 -

Naser M.
1971
Hyperholomorphic functions
Silberian Math. J.
12
959 -
968

Nôno K.
1983
Hyperholomorphic functions of a quaternion variable
Bull. Fukuoka Univ. Ed.
32
21 -
37

Citing 'CHARACTERIZATION OF A REGULAR FUNCTION WITH VALUES IN DUAL QUATERNIONS
'

@article{ SHGHCX_2015_v22n1_65}
,title={CHARACTERIZATION OF A REGULAR FUNCTION WITH VALUES IN DUAL QUATERNIONS}
,volume={1}
, url={http://dx.doi.org/10.7468/jksmeb.2015.22.1.65}, DOI={10.7468/jksmeb.2015.22.1.65}
, number= {1}
, journal={The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={KIM, JI EUN
and
SHON, KWANG HO}
, year={2015}
, month={Feb}