CHARACTERIZATION OF A REGULAR FUNCTION WITH VALUES IN DUAL QUATERNIONS
CHARACTERIZATION OF A REGULAR FUNCTION WITH VALUES IN DUAL QUATERNIONS
The Pure and Applied Mathematics. 2015. Feb, 22(1): 65-74
Copyright © 2015, Korean Society of Mathematical Education
• Received : October 31, 2014
• Accepted : January 10, 2015
• Published : February 28, 2015
PDF
e-PUB
PPT
Export by style
Article
Author
Metrics
Cited by
TagCloud
JI EUN, KIM
DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY, BUSAN 609-735 REPUBLIC OF KOREAEmail address:jeunkim@pusan.ac.kr
KWANG HO, SHON
DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY, BUSAN 609-735 REPUBLIC OF KOREAEmail address:khshon@pusan.ac.kr

Abstract
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.
Keywords
1. INTRODUCTION
Let 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
PPT Slide
Lager Image
of matrix algebra
PPT Slide
Lager Image
. 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.
2. PRELIMINARIES
The field T of quaternions
PPT Slide
Lager Image
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 :
PPT Slide
Lager Image
The element e 0 = 1 is the identity of T . Identifying the element e 1 with the imaginary unit
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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.
3. DUAL QUATERNIONS
The algebra
PPT Slide
Lager Image
where
PPT Slide
Lager Image
, is a non-commutative subalgebra of
PPT Slide
Lager Image
.
We define that the dual quaternionic multiplication of two dual quaternions
PPT Slide
Lager Image
and
PPT Slide
Lager Image
is given by
PPT Slide
Lager Image
The dual quaternionic conjugate Z * of Z is
PPT Slide
Lager Image
Then the modulus | Z | and the inverse Z −1 of Z in
PPT Slide
Lager Image
are defined by the following :
PPT Slide
Lager Image
and
PPT Slide
Lager Image
By using the multiplication of
PPT Slide
Lager Image
, the power of Z is for
PPT Slide
Lager Image
,
PPT Slide
Lager Image
and the division of two
PPT Slide
Lager Image
can be computed as
PPT Slide
Lager Image
Since
PPT Slide
Lager Image
and
PPT Slide
Lager Image
are real variables, it can be written by
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
We use the following differential operators :
PPT Slide
Lager Image
PPT Slide
Lager Image
where
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
( k = 1, 2) are usual complex differential operations.
The Laplacian operator is
PPT Slide
Lager Image
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
PPT Slide
Lager Image
:
PPT Slide
Lager Image
where uj = uj ( x 0 , x 1 , x 2 , x 3 , y 0 , y 1 , y 2 , y 3 ) and vj = vj ( 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:
PPT Slide
Lager Image
where
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Also, it is equivalent to
PPT Slide
Lager Image
The above system is called a dual quaternion Cauchy-Riemann system in dual quaternions.
Let ­ Ω be an open subset of
PPT Slide
Lager Image
, for Z 0 = z 0 + εω 0 ∈ Ω­,
PPT Slide
Lager Image
is called a dual-quaternion function in
PPT Slide
Lager Image
.
Definition 3.4. A function F is said to be continuous at Z 0 = z 0 + εω 0 if
PPT Slide
Lager Image
where the limit has
PPT Slide
Lager Image
Definition 3.5. The dual quaternion function F is said to be differentiable in dual quaternions if the limit
PPT Slide
Lager Image
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
PPT Slide
Lager Image
PPT Slide
Lager Image
where
PPT Slide
Lager Image
is a constant in a domain of f (see [2 , 11] ). Since the equation (3.2) is equivalent to Dzf , we can express
PPT Slide
Lager Image
. Hence, by the representations of DF and properties of limit, calculating the division for
PPT Slide
Lager Image
PPT Slide
Lager Image
Therefore, we can represent
PPT Slide
Lager Image
Theorem 3.7. Let F = f + εg be a dual quaternion function in ­
PPT Slide
Lager Image
. If F satisfies the equation Df = 0, then the derivative of F satisfies the following equation:
PPT Slide
Lager Image
Proof. By the division of dual quaternions, we have
PPT Slide
Lager Image
Then, the limit
PPT Slide
Lager Image
exists if and only if
PPT Slide
Lager Image
has two cases to deal with
Case 1)
PPT Slide
Lager Image
If
PPT Slide
Lager Image
then the limit exists and the derivative can be written by
PPT Slide
Lager Image
Case 2)
PPT Slide
Lager Image
If
PPT Slide
Lager Image
then the limit exists and the derivative can be written by
PPT Slide
Lager Image
Therefore, the equation
PPT Slide
Lager Image
is obtained.
PPT Slide
Lager Image
Theorem 3.8. Let F = f + εg be a dual quaternion function in ­
PPT Slide
Lager Image
. 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:
PPT Slide
Lager Image
Proof. From the proof of Theorem 3.7, we have
PPT Slide
Lager Image
Since F satisfies a dual quaternion Cauchy-Riemann system (3.1), we have
PPT Slide
Lager Image
Therefore, since
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
PPT Slide
Lager Image
References
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 Hanoi, Vietnam, Hue Univ. 14 127 - 155
Kim J.E. , Lim S.J. , Shon K.H. 2013 Taylor series of functions with values in dual quaternion J. Korean Soc. Math. Educ., Ser. B, Pure Appl. Math. 20 (4) 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 60 1 - 9
Koriyama H. , Mae H. , Nôno K. 2011 On regularities of octonionic functions and holomorphic mappings Bull. Fukuoka Univ. Ed. part III 60 11 - 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