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TAYLOR SERIES OF FUNCTIONS WITH VALUES IN DUAL QUATERNION
TAYLOR SERIES OF FUNCTIONS WITH VALUES IN DUAL QUATERNION
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2013. Oct, 20(4): 251-258
Copyright © 2013, Korean Society of Mathematical Education
  • Received : July 08, 2013
  • Accepted : November 07, 2013
  • Published : October 30, 2013
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About the Authors
지은 김
DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY, BUSAN 609-735, KOREAEmail address:jeunkim@pusan.ac.kr
수진 임
DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY, BUSAN 609-735, KOREAEmail address:sjlim@pusan.ac.kr
강호 손
DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY, BUSAN 609-735, KOREAEmail address:khshon@pusan.ac.kr

Abstract
We define an ε-regular function in dual quaternions. From the properties of ε-regular functions, we represent the Taylor series of ε-regular functions with values in dual quaternions.
Keywords
1. INTRODUCTION
Fueter [2] and Naser [8] have studied properties of quaternionic differential equations as a generalization of the extended Cauchy-Riemann equations in the complex holomorphic function theory and Nôno [9 , 10 , 11] has given a definition of regular functions over the quaternion field
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identified with
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In 1979, Sudbery [15] developed quaternionic regular function theories. By using a generalization of the Cauchy-Riemann equation, Ryan [12 , 13] has developed regular function theories on complex Cliffrd algebra of quaternion valued functions.
In 1873, Clifford [1] originally conceived the algebra of dual numbers. Dual algebra has been often used for closed form solutions in the field of displacement analysis. Kotelnikov [6] and Study [14] developed dual vectors and dual quaternions for use in the application of mechanics and realized that this associative algebra was ideal for describing the group of motions of three-dimensional spaces.
In 2011, Koriyama, Mae and Nôno [5] investigated hyperholomorphic functions and holomorphic functions in quaternion analysis. In 2012, Gotô and Nôno [3] researched regular functions with values in a commutative subalgebra of matrix algebra in four real dimension and we [7] obtained regularities of functions with values in subalgebra of matrix algebras in complex n -dimensional.
In this paper, we introduce the dual quaternion numbers and give some properties of ε-regular functions in dual quaternions by using the associated Pauli matrices. We give the notation of the derivative for functions with values in dual quaternions and obtain the representation of the Taylor series of ε-regular functions.
2. PRELIMINARY
A dual quaternion is an ordered pair of quaternions and is constructed from eight base elements e 0 , e 1 , e 2 , e 3 , ε, e 1 ε, e 2 ε and e 3 ε. We consider the associated Pauli matrices where
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. And, we let the dual quaternion identity which is a nonzero and satisfy 0ε = ε0 = 0, 1ε = ε1 = ε, ε 2 = 0 and where
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,
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and
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is a dual quaternion component of
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The element e 0 is the identity, the element ε is the dual identity of
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and the element e 1 identifies the imaginary unit
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in the ℂ-field of complex numbers. We can identify
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with
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The dual quaternionic conjugation z of z , the absolute value | z | of z and an inverse z −1 of z in
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are defined, respectively, by where
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and
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.
Let Ω be an open subset of
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and the dual quaternion function f : Ω → satisfy where
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and
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are real-valued functions.
We use the following two dual quaternion differential operators which are defined as and where Then we have where
Definition 2.1. Let Ω be an open set in
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. A function f ( z ) is said to be ε-regular in Ω if the following two conditions are satisfied:
(a) fj ( j = 0,1,2,3) are continuously diffrential functions in Ω, and
(b) D f ( z ) = 0 in Ω.
3. TAYLOR SERIES OF DUAL QUATERNION FUNCTIONS
We define the derivative f '( z ) of f ( z ) by the following: f'(z) := Df(z)
Lemma 3.1. Let Ω be a domain in
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and f ( z ) be a holomorphic mapping and ε-regular defined in Ω. Then , Proof . Since f ( z ) is an ε-regular function in Ω, we have where From we can get
Theorem 3.2. Let f ( z ) be a homogeneous polynomial of degree m with respect to the variables ξ and ξ?. If f(z) is a holomorphic and ε-regular function in
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then we have
Proof . Since f ( z ) is a homogeneous polynomial, we have
Then f'(z) = m-1+εm(m - 1)ξm-2ξ*f'(z)z = m+εm2ξm-1ξ*.
Thus,
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. And f''(z) = m(m - 1)ξm-2+εm(m - 1)(m - 2)ξm-3ξ*f''(z)z = m(m - 1)ξm-1+εm(m - 1)2ξm-2ξ*.
Thus,
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Repeating the above calculation, we have
Theorem 3.3. Let Ω be a domain in
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Let f ( z ) be a holomorphic and ε-regular function in Ω and α ∈ Ω. Then there exists a neighborhood U α of α such that where
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.
Proof . From substituting a dual number into Taylor series, we have
By Theorem 3.2 and zm = ξm + ε (( m − 1) ξ m−1 ξ + ξ m−2 ξ ), we have
Remark 3.4. Let Ω be a domain in
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. If g 0 ( z ) is a holomorphic function with value in quaternions, then there exists a function g 1 ( z ) with value in quaternions such that f(z) = g0(z)+εg1(z) is ε-regular in Ω.
By the results of Kenwright [4] , f(z) = f(ξ)+εf′(ξ)ξ.
We put g 0 ( z ) = f ( ξ ), then Dg 0 ( z ) = f ′( ξ ). Hence we put g 1 ( z ) = Dg 0 ( z ) ξ , we have f(z) = g0(z)+εg1(z).
Example 3.5. Let Ω be a domain in
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If g 0 = sin(nζ), then g 0 = n cos( ), n ∈ℤ. Thus, there exists a function g 1 = n cos( ) ξ * such that f(z) = g0(z)+εg1(z). is ε-regular in Ω.
This work was supported by a 2-Year Research Grant of Pusan National University.
References
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Gottô S. , Ntôno K. 2012 Regular functions with values in a Commutative subalgebra ℂ(ℂ) of Matrix algebra M(4;4;ℝ), part III Bull. Fukuoka Univ. Ed. 61 9 - 15
Kenwright B. 2012 A beginners guide to dual-quaternions: What they are, How they work, and How to use them for 3D character hierarchies The 20th International Conf. on Computer Graphics, Visualization and Computer Vision June 26-28 1 - 10
Koriyama H. , Mae H. , Nôno K. 2011 Hyperholomorphic Functions and Holomorphic functions in Quaternionic Analysis, part III Bull. Fukuoka Univ. Ed. 60 1 - 9
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Study E. 1903 Geometrie der Dynamen Teubner Leipzig
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