# Dual Quaternion Functions and Its Applications

Authors: Su Jin Lim; Kwang Ho Shon
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A dual quaternion is associated with two quaternions that have basis elementse0, e1, e2, e3, and ε. Dual
numbers are often written in the form z=ζ+εζ*, where ε is the dual identity and has the properties ε2=0(ε≠0). We research the properties of some regular functions with values in dual quaternion and give applications
of the extension problem for dual quaternion functions.


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## Body

## 1. Introduction

Let𝒯 be the quaternion algebra constructed over a real anti-Euclidean quadratic four-dimensional vector space. Brackx [1], Deavours [2], and Sudbery [3] researched properties of theories of a quaternion function. Naser [4] gave properties of hyperholomorphic functions, and Nôno [5, 6] gave properties of various hyperholomorphic functions. They obtained basic theorems such as Cauchy Theorem, Morera’s Theorem, and Cauchy Integral Formula with respect to Clifford analysis. Also, we [7–10] have investigated certain properties of hyperholomorphic functions and some regular functions in Clifford analysis.A dual quaternion algebra𝒟ℋ is an ordered pair of quaternions and is constructed from real eight-dimensional vector spaces. A dual quaternion can be represented in the form z=ζ+εζ*, where ζ and ζ* are ordinary quaternions and ε is the dual symbol. The quaternion can represent only rotation, while the dual quaternion can do both rotation and translation. So, the dual quaternion is used in applications to 3D computer graphics, robotics, and computer vision. Kenwright [11] gave characteristics of dual quaternions; Pennestrì and Stefanelli [12] researched some properties using dual, and Kula and Yayli [13] investigated dual split quaternions and screw motion in Minkowski 3-space.Son [14–16] gave the extension problem for the solutions of partial differential equations in Rn and it is generalized for the solutions of the Riesz system. In this paper, we give some regular functions with values in dual quaternions and research the extension problem for regular functions with values in dual quaternions. Also, we give some applications for these problems.


## 2. Preliminaries

We consider associated Pauli matrices(1)e0=(1001),e1=(i00-i),e2=(01-10),e3=(0ii0).
Then the associated Pauli matrices satisfy the triple rule as follows:
(2)ej2=-1,ejek+ekej=-δjk(j,k=1,2,3),
where δjk is Kronecker delta. And we let the dual symbol
(3)ε=(0100)
be a nonzero and satisfy 0ε=ε0=0, 1ε=ε1=ε, ε2=0. The element e0 is the identity and the element ε is the dual identity of 𝒯.The dual quaternion algebra𝒟ℋ is a noncommutative and associative one of the quaternion algebra. Then
(4)𝒟ℋ∶={z=∑j=03‍(ejxj+ejxj*ε)∣xj,xj*∈R𝒟ℋ∶=12z=∣xj,xj*R.∈R∑j=03‍(j=0,1,2,3)}={z=ζ+ζ*ε∣ζ,ζ*∈𝒯}≅𝒯×𝒯,
where ζ=∑j=03‍ejxj, ζ*=∑j=03‍ejxj*, and xj* is a dual quaternion component of xj. We can identify 𝒟ℋ with C4. The numbers of the skew field 𝒟ℋ of dual quaternions are
(5)z=∑j=03‍(ejxj+ejxj*ε)=∑j=03‍ejξj=(ξ0+iξ1ξ2+iξ3-ξ2+iξ3ξ0-iξ1),w=∑j=03‍(ejyj+ejyj*ε)=∑j=03‍ejηj=(η0+iη1η2+iη3-η2+iη3η0-iη1),
where ξj=xj+εxj* and ηj=yj+εyj*(j=0,1,2,3). The dual quaternion conjugate z* of z is
(6)z*=∑j=03‍(ej¯xj+ej¯xj*ε)=∑j=03‍ej¯ξj=(ξ0-iξ1-ξ2-iξ3ξ2-iξ3ξ0+iξ1),
where ej¯=-ej. The absolute value |z| of z and the inverse z-1 of z are, respectively,
(7)|z|=zz*=∑j=03‍ξj2,z-1=z*|z|2(z≠0).LetΩ be an open subset of C2×C2 and let the dual quaternion function
(8)f:Ω⟶𝒟ℋ
satisfy
(9)z∈Ω⟼f(z)=∑j=03‍ejfj(ζ,ζ*)=(f0(ζ,ζ*)+if1(ζ,ζ*)f2(ζ,ζ*)+if3(ζ,ζ*)-f2(ζ,ζ*)+if3(ζ,ζ*)f0(ζ,ζ*)-if1(ζ,ζ*))∈𝒟ℋ,
where fj(ζ,ζ*)=uj(ζ,ζ*)+εuj*(ζ,ζ*) and uj, uj*  (j=0,1,2,3) are real-valued functions.We use the following dual quaternion differential operators in𝒟ℋ:
(10)D=∑j=03‍ej¯∂∂qj=(∂∂q0-i∂∂q1-∂∂q2-i∂∂q3∂∂q2-i∂∂q3∂∂q0+i∂∂q1),
and the dual quaternion conjugates differential operators
(11)D*=∑j=03‍ej∂∂qj=(∂∂q0+i∂∂q1∂∂q2+i∂∂q3-∂∂q2+i∂∂q3∂∂q0-i∂∂q1),
where ∂/∂qj=∂/∂xj+ε(∂/∂xj*)(j=0,1,2,3) and qj=xj+(1/ε)xj*. Then we have
(12)DD*=∑j=03‍∂2∂qj2=Δq.Definition 1.
LetΩ be an open set in C2×C2. 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 differential functions in Ω,(b)
D*f(z)=0 in Ω.Definition 2.
LetΩ be an open set in C2×C2. A function f(z) is said to be ε-biregular in Ω if the following two conditions are satisfied:(a)
fj(j=0,1,2,3) are continuously differential functions in Ω,(b)
D*f(z)=0 and f(z)D*=0 in Ω.The operators act for a functionf(z) on 𝒟ℋ as follows:
(13)D*f(z)=(∑j=03‍ej∂∂qj)(∑j=03‍ejfj)=(D1*D2*D3*D4*),
where
(14)D1*=(∂f0∂q0-∂f1∂q1-∂f2∂q2-∂f3∂q3)+i(∂f1∂q0+∂f0∂q1+∂f3∂q2-∂f2∂q3),D2*=(∂f2∂q0-∂f3∂q1+∂f0∂q2+∂f1∂q3)+i(∂f3∂q0+∂f2∂q1-∂f1∂q2+∂f0∂q3),D3*=(-∂f2∂q0+∂f3∂q1-∂f0∂q2-∂f1∂q3)+i(∂f3∂q0+∂f2∂q1-∂f1∂q2+∂f0∂q3),D4*=(∂f0∂q0-∂f1∂q1-∂f2∂q2-∂f3∂q3)+i(-∂f1∂q0-∂f0∂q1-∂f3∂q2+∂f2∂q3),f(z)D*=(∑j=03‍ejfj)(∑j=03‍ej∂∂qj)=(D5*D6*D7*D8*),
where
(15)D5*=(∂f0∂q0-∂f1∂q1-∂f2∂q2-∂f3∂q3)+i(∂f1∂q0+∂f0∂q1-∂f3∂q2+∂f2∂q3),D6*=(∂f2∂q0+∂f3∂q1+∂f0∂q2-∂f1∂q3)+i(∂f3∂q0-∂f2∂q1+∂f1∂q2+∂f0∂q3),D7*=(-∂f2∂q0-∂f3∂q1-∂f0∂q2+∂f1∂q3)+i(∂f3∂q0-∂f2∂q1+∂f1∂q2+∂f0∂q3),D8*=(∂f0∂q0-∂f1∂q1-∂f2∂q2-∂f3∂q3)+i(-∂f1∂q0-∂f0∂q1+∂f3∂q2-∂f2∂q3).Remark 3.
Equations (b) of Definition2 are equivalent to the following system:
(16)∂f0∂q0-∂f1∂q1-∂f2∂q2-∂f3∂q3=0,∂f1∂q0=-∂f0∂q1,∂f3∂q2=∂f2∂q3,∂f2∂q0=-∂f0∂q2,∂f3∂q1=∂f1∂q3,∂f3∂q0=-∂f0∂q3,∂f2∂q1=∂f1∂q2.


## 3. Extension Problem for the Dual Quaternion Functions

Definition 4.
LetΩ be a domain in Cn×Cn  (n≥1). A function f(z)=∑j=0n-1‍ejfj(z) is said to be regular in Ω if
(17)D~f(z)=0,
where D~=∑j=0n-1‍ej(∂/∂qj) on Ω.Theorem 5 (uniqueness theorem for regular functions).
If two regular functionsf and g in a domain Ω⊂Cn×Cn  (n≥1) and coincide on a nonempty open set G⊂Ω, then f≡g in Ω.Rocha-Chávez et al. [17] obtained the following remark.Remark 6.
For a regular functionf in the domain Ω⊂Cn×Cn(n≥1) and a bounded domain G with smooth boundary bG, such that G¯⊂Ω, one has
(18)f(z)=1an∫bG‍ζ¯-z¯|ζ-z|ndσζf(ζ),z∈G,
with an the area of the unit sphere in Cn and dσζ a Clifford algebra valued differential form of order n-1.LetΩ=Ω1×Ω2 be a domain in C4×Cn-4(n≥5) where Ω1 is a domain in C4(ξ0,ξ1,ξ2,ξ3) and Ω2 is a domain in Cn-4(ξ4,ξ5,…,ξn-1). Let U be an open connected neighborhood of bΩ.Proposition 7.
Iff(z) is a regular function in U⊂C4×Cn-4(n≥5) which satisfies the condition
(19)D*f(z)=0,
then f(z) can be extended continuously to a regular function in the whole domain of Ω. That is, there exists a regular function f~(z) in Ω such that f~(z)=f(z) in U.Proof.
By Remark6 and the proof of the main extension theorem of Son [15], it is proved.We consider the system of an extension of the system (16)
(20)∂f0∂q0-∂f1∂q1-⋯-∂fn-1∂qn-1=0,∂fj∂q0=-∂f0∂qj,∂fl∂qk=∂fk∂ql=0∂fk∂ql(j,k,l=1,…,n-1),
where f(z)={f0(z),f1(z),…,fn-1(z)} are the unknown functions.By using the same technique as in Son [15], we have the following theorem.Theorem 8.
Letf(z)={f0(z),f1(z),…,fn-1(z)} be a given 𝒞2-solution of the system (20) in U⊂C4×Cn-4(n≥5), which satisfies the system (16) in Remark 3. If the functions f4(z),f5(z),…,fn-1(z) depend only on ξ0,ξ1,ξ2,ξ3, and U is an open neighborhood of the boundary of the domain Ω⊂C4×Cn-4(n≥5), then f(z) can be extended to a solution of the system (20) in the whole domain of Ω.Proof.
Let the functionϕ(z) with values in Clifford algebra be defined by
(21)ϕ(z)=∑j=0n-1‍fjej.
Then we have D~ϕ(z)=0 and D*ϕ(z)=0. By Proposition 7, the result follows.We consider the following system:(22)∑j=0n-1‍∑k=0m-1‍Hjk(P)(z)∂fj∂qk=0(P=1,…,p),
where Hjk(P)(z) are holomorphic functions and f={f0,f1,…,fn-1} are the unknown functions of the system (22).LetΩ be an open set in Cn  (n≥2) and let K be a compact subset of Ω such that Ω∖K is simply connected. We consider the system
(23)∑j=0n-1‍∑k=0m-1‍Hjk(P)(z)∂ωj∂qk=φ(z)(P)(P=1,…,p),
where φ(z)(P)∈𝒞∞(Ω).By using the same technique as in Son [16], we have the following theorem and example.Theorem 9.
If everyφ={φ(1),…,φ(p)}∈𝒞∞(Ω) in the inhomogeneous system (23) has a solution
(24)ω={ω0,…,ωn-1}∈𝒞∞(Ω),
then every solution f of (22) given in Ω∖K can be extended to a solution of this system (23) in the whole domain of Ω.Proof.
This result follows from the theorem in [18, page 30].Example 10.
We give an application of Theorem9 to the system (20) and recall the system (16) as follows:
(25)∂f0∂q0-∂f1∂q1-⋯-∂fn-1∂qn-1=0,∂fj∂q0+∂f0∂qj=0,∂fl∂qk-∂fk∂ql=0-∂fk∂ql0=01(j,k,l=1,…,n-1).Assume that(26)∂fn-1∂qn-1=0.
Then we have the following form:
(27)∂f0∂q0-∂f1∂q1-⋯-∂fn-2∂qn-2=0,∂fj∂q0+∂f0∂qj=0,∂fl∂qk-∂fk∂ql=0-∂fk∂ql=001(j,k,l=1,…,n-2).
The corresponding inhomogeneous system of (27) has the following form:
(28)∂ω0∂q0-∂ω1∂q1-⋯-∂ωn-2∂qn-2=φ,∂ωj∂q0+∂ω0∂qj=φj,0,∂ωj∂qk-∂ωk∂qj=φj,k,∂ωk∂q0+∂ω0∂qk=φk,0,∂ωj∂ql-∂ωl∂qj=φj,l,∂ωk∂ql-∂ωl∂qk=φk,l-∂ωn-2∂qn-2(j,k,l=1,…,n-2),
where φ,φj,0,φj,k,φk,0,φj,l,φk,l∈𝒞∞(Ω). Then we can get the system from (28) as
(29)∂ωk∂qj=∂ωj∂qk-φj,k,∂ωk∂q0=-∂ω0∂qk+φk,0,∂ωl∂qj=∂ωj∂ql-φj,l,∂ωl∂qk=∂ωk∂ql-φk,l,(30)∂2ωk∂qj∂q0+∂φj,k∂q0=-∂2ω0∂qj∂qk+∂φj,0∂qk,∂2ωj∂ql∂qk-∂φj,l∂qk=∂2ωk∂qj∂ql-∂φk,l∂qj.
From (29), we have
(31)∂∂qj(-∂ω0∂qk+φk,0)+∂φj,k∂q0=-∂2ω0∂qj∂qk+∂φj,0∂qk,∂∂ql(∂ωk∂qj+φj,k)-∂φj,l∂qk=∂2ωk∂qj∂ql-∂φk,l∂qj.
Thus, we can have the system
(32)∂φk,0∂qj+∂φj,k∂q0=∂φj,0∂qk,∂φj,k∂ql-∂φj,l∂qk=-∂φk,l∂qj.
We let
(33)ω~=∑j=0n-2‍ejωj,z~=∑j=0n-2‍ej¯ξj.
The system (29) has the form
(34)D~ω~=F(z~,ξn-1),∂ω0∂qk+∂ωk∂q0=φk,0,(35)∂ωj∂ql-∂ωl∂qj=φj,l,∂ωk∂ql-∂ωl∂qk=φk,l,
where
(36)F(z~,ξn-1)=φ+∑j=1n-2‍ejφj,0+(n2-1)∑α=1n-2‍∑j,k=1j,k≠αn-2eαφj,k.
We put
(37)F1:=φ+∑j=1n-2‍ejφj,0,F2:=(n2-1)∑α=1n-2‍∑j,k=1j,k≠αn-2eαφj,k.
From the system (28), we have
(38)ω0=∫‍(φ+∂ω1∂q1+∂ω2∂q2+⋯+∂ωn-2∂qn-2)dq0,ωk=∫‍(φk,0-∂ω0∂qk)dq0(k=1,…,n-2).
By the systems (29) and (38), we get
(39)ω~=∫‍Fdq0+∫‍∑j=1n-2‍∂ωj∂qjdq0-∫‍∑j=1n-2‍ej∂ω0∂qjdq0-∫‍F2dq0=∫‍Fdq0-∫‍∑j=1n-2‍ej∂ω~∂qjdq0.
By Cauchy Integral Formula,
(40)F(Z,ξn-1)=1(2πi)n∫‍F(Z,ξn-1)Z-ζdZ,
where Z=∑k=0n-2‍ekZk. Thus we have
(41)ω~=1(2πi)n∬F(Z,ξn-1)Z-ζdZ0dZ,
and ω~=0 when |ξn-1| is large enough. Also, {ω~,0}={ω0,ω1,…,ωn-2,0} is a solution of the system (22) outside a compact set K of Ω. From Theorem 5, it follows that ω~=0 is outside the compact set K of Ω or ω∈𝒞∞(Ω). That is, ω0,ω1,…,ωn-2∈𝒞∞(Ω). It follows from the system (35) that
(42)ωn-1=∫‍(∂ωm-1∂qn-1-φm-1,n-1)dqm-1.
Since ωm-1∈𝒞∞(Ω), we get ∂ωm-1/∂qn-1∈𝒞∞(Ω). We can choose ωn-1∈𝒞∞(Ω) which satisfy the system (35). From (42), we find that
(43)∂ωn-1∂qP=∫‍(∂2ωm-1∂qP∂qn-1-∂φm-1,n-1∂qP)dqm-1=∫‍(∂∂qn-1(∂ωP∂qm-1)+∂φm-1,P∂qn-1-∂φm-1,n-1∂qP)dqm-1=∫‍(∂∂qn-1(∂ωP∂qm-1)-∂φP,n-1∂qm-1)dqm-1=∂ωP∂qn-1-φP,n-1.
Hence, ωn-1 satisfies the system (29). This means that the function ω=(ω~,ωn-1)=(ω0,…,ωn-1) is a solution of the system (29) and ω∈𝒞∞(Ω).



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