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

- Received : July 08, 2013
- Accepted : November 07, 2013
- Published : October 30, 2013

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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.
identiﬁed with
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.
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
. And, we let the dual quaternion identity
which is a nonzero and satisfy 0ε = ε0 = 0, 1ε = ε1 = ε, ε
^{2}
= 0 and
where
,
and
is a dual quaternion component of
The element
e
_{0}
is the identity, the element ε is the dual identity of
and the element
e
_{1}
identiﬁes the imaginary unit
in the ℂ-ﬁeld of complex numbers. We can identify
with
The dual quaternionic conjugation
z
^{∗}
of
z
, the absolute value |
z
| of
z
and an inverse
z
^{−1}
of
z
in
are deﬁned, respectively, by
where
and
.
Let Ω be an open subset of
and the dual quaternion function
f : Ω →
satisfy
where
and
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
. A function
f
(
z
) is said to be ε-regular in Ω if the following two conditions are satisﬁed:
(a)
f_{j}
(
j
= 0,1,2,3) are continuously diffrential functions in Ω, and
(b)
D
^{∗}
f
(
z
) = 0 in Ω.
f
'(
z
) of
f
(
z
) by the following:
f '(z ) := Df (z )
Lemma 3.1.
Let
Ω
be a domain in
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
then we have
Proof
. Since
f
(
z
) is a homogeneous polynomial, we have
Then
f' (z ) = mξ^{m-1}+εm (m - 1)ξ^{m-2}ξ* f' (z )z = mξ^{m}+εm^{2}ξ^{m-1}ξ*.
Thus,
. And
f ''(z ) = m (m - 1 )ξ^{m-2} +εm (m - 1)(m - 2)ξ^{m-3}ξ *f ''(z )z = m (m - 1 )ξ^{m-1} +εm (m - 1)^{2}ξ^{m-2}ξ *.
Thus,
Repeating the above calculation, we have
Theorem 3.3.
Let
Ω
be a domain in
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
.
Proof
. From substituting a dual number into Taylor series, we have
By Theorem 3.2 and
z^{m}
=
ξ^{m}
+
ε
((
m
− 1)
ξ
^{m−1}
ξ
^{∗}
+
ξ
^{m−2}
ξ
^{∗}
), we have
Remark 3.4.
Let Ω be a domain in
. 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 ) = g _{0}(z )+εg _{1}(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 ) = g _{0}(z )+εg _{1}(z ).
Example 3.5.
Let Ω be a domain in
If
g
_{0 }
= sin(nζ), then
g
′
_{0}
=
n
cos(
nξ
),
n
∈ℤ. Thus, there exists a function
g
_{1}
=
n
cos(
nξ
)
ξ
* such that
f (z ) = g _{0}(z )+εg _{1}(z ).
is ε-regular in Ω.
* This work was supported by a 2-Year Research Grant of Pusan National University.

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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2. PRELIMINARY

A dual quaternion is an ordered pair of quaternions and is constructed from eight base elements
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3. TAYLOR SERIES OF DUAL QUATERNION FUNCTIONS

We deﬁne the derivative
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Clifford W.K.
1873
Preliminary sketch of bi-quaternions
Proc. London Math. Soc.
4
381 -
395

Fueter R.
1934
Die Fuktionentheorie der Defferentialgeleichungen ∆u = 0 und ∆∆u = 0 mit vier reellen Variablen
Comment. Math. Helv.
7
307 -
330
** DOI : 10.1007/BF01292723**

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

Kotelnikov A.P.
1895
Screw calculus and some applications to geometry and mechanics
Annals Imperial Univ.
Kazan

Lim S.J.
,
Shon K.H.
2013
Regularity of functions with values in a non-commutative algebra of complex matrix algebras
Sci. China Math.

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

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

1986
Characterization of domains of holomorphy by the existence of hyper-conjugate harmonic functions
Rev. Roumaine Math. Pures Appl.
31
(2)
159 -
161

1987
Domains of Hyperholomorphic in ℂ2 × ℂ2
Bull. Fukuoka Univ. Ed.
36
1 -
9

Ryan J.
1983
Complex Clifford Analysis
Complex Variables Theory Appl.
1
119 -
149

1983
Special functions and relations within complex Clifford analysis I
Complex Variables Theory Appl.
2
177 -
198
** DOI : 10.1080/17476938308814041**

Study E.
1903
Geometrie der Dynamen
Teubner
Leipzig

Sudbery A.
1979
Quaternionic analysis
Math. Proc. Camb. Phil. Soc.
85
199 -
225
** DOI : 10.1017/S0305004100055638**

Citing 'TAYLOR SERIES OF FUNCTIONS WITH VALUES IN DUAL QUATERNION
'

@article{ SHGHCX_2013_v20n4_251}
,title={TAYLOR SERIES OF FUNCTIONS WITH VALUES IN DUAL QUATERNION}
,volume={4}
, url={http://dx.doi.org/10.7468/jksmeb.2013.20.4.251}, DOI={10.7468/jksmeb.2013.20.4.251}
, number= {4}
, journal={Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={김, 지은
and
임, 수진
and
손, 강호}
, year={2013}
, month={Oct}