• Views 895
• Citations 5
• ePub 26
• PDF 728
`Journal of Applied MathematicsVolume 2013, Article ID 583813, 6 pageshttp://dx.doi.org/10.1155/2013/583813`
Research Article

## Dual Quaternion Functions and Its Applications

Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea

Received 7 June 2013; Accepted 23 July 2013

Copyright © 2013 Su Jin Lim and Kwang Ho Shon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A dual quaternion is associated with two quaternions that have basis elements , , , , and . Dual numbers are often written in the form , where is the dual identity and has the properties . We research the properties of some regular functions with values in dual quaternion and give applications of the extension problem for dual quaternion functions.

#### 1. Introduction

Let 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 [710] have investigated certain properties of hyperholomorphic functions and some regular functions in Clifford analysis.

A dual quaternion algebra is an ordered pair of quaternions and is constructed from real eight-dimensional vector spaces. A dual quaternion can be represented in the form , where and are ordinary quaternions and 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 [1416] gave the extension problem for the solutions of partial differential equations in 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 Then the associated Pauli matrices satisfy the triple rule as follows: where is Kronecker delta. And we let the dual symbol be a nonzero and satisfy , , . The element is the identity and the element is the dual identity of .

The dual quaternion algebra is a noncommutative and associative one of the quaternion algebra. Then where , , and is a dual quaternion component of . We can identify with . The numbers of the skew field of dual quaternions are where and . The dual quaternion conjugate of is where . The absolute value of and the inverse of are, respectively,

Let be an open subset of and let the dual quaternion function satisfy where and ,    are real-valued functions.

We use the following dual quaternion differential operators in : and the dual quaternion conjugates differential operators where and . Then we have

Definition 1. Let be an open set in . A function is said to be -regular in if the following two conditions are satisfied:(a) are continuously differential functions in ,(b) in .

Definition 2. Let be an open set in . A function is said to be -biregular in if the following two conditions are satisfied:(a) are continuously differential functions in , (b) and in .

The operators act for a function on as follows: where where

Remark 3. Equations (b) of Definition 2 are equivalent to the following system:

#### 3. Extension Problem for the Dual Quaternion Functions

Definition 4. Let be a domain in   . A function is said to be regular in if where on .

Theorem 5 (uniqueness theorem for regular functions). If two regular functions and in a domain    and coincide on a nonempty open set , then in .

Rocha-Chávez et al. [17] obtained the following remark.

Remark 6. For a regular function in the domain and a bounded domain with smooth boundary , such that , one has with the area of the unit sphere in and a Clifford algebra valued differential form of order .

Let be a domain in where is a domain in and is a domain in . Let be an open connected neighborhood of .

Proposition 7. If is a regular function in which satisfies the condition then can be extended continuously to a regular function in the whole domain of . That is, there exists a regular function in such that in .

Proof. By Remark 6 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) where are the unknown functions.

By using the same technique as in Son [15], we have the following theorem.

Theorem 8. Let be a given -solution of the system (20) in , which satisfies the system (16) in Remark 3. If the functions depend only on , and is an open neighborhood of the boundary of the domain , then can be extended to a solution of the system (20) in the whole domain of .

Proof. Let the function with values in Clifford algebra be defined by Then we have and . By Proposition 7, the result follows.

We consider the following system: where are holomorphic functions and are the unknown functions of the system (22).

Let be an open set in    and let be a compact subset of such that is simply connected. We consider the system where .

By using the same technique as in Son [16], we have the following theorem and example.

Theorem 9. If every in the inhomogeneous system (23) has a solution then every solution of (22) given in can be extended to a solution of this system (23) in the whole domain of .

Proof. This result follows from the theorem in [18, page 30].

Example 10. We give an application of Theorem 9 to the system (20) and recall the system (16) as follows:

Assume that Then we have the following form: The corresponding inhomogeneous system of (27) has the following form: where . Then we can get the system from (28) as From (29), we have Thus, we can have the system We let The system (29) has the form where We put From the system (28), we have By the systems (29) and (38), we get By Cauchy Integral Formula, where . Thus we have and when is large enough. Also, is a solution of the system (22) outside a compact set of . From Theorem 5, it follows that is outside the compact set of or . That is, . It follows from the system (35) that Since , we get . We can choose which satisfy the system (35). From (42), we find that Hence, satisfies the system (29). This means that the function is a solution of the system (29) and .

#### Acknowledgment

The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2013R1A1A2008978).

#### References

1. F. Brackx, “On (k)-monogenic functions of a quaternion variable,” in Function Theoretic Methods in Differential Equations, vol. 8 of Research Notes in Mathematics, pp. 22–44, 1876.
2. C. A. Deavours, “The quaternion calculus,” The American Mathematical Monthly, vol. 80, pp. 995–1008, 1973.
3. A. Sudbery, “Quaternionic analysis,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 85, no. 2, pp. 199–224, 1979.
4. M. Naser, “Hyperholomorphic functions,” Silberian Mathematical Journal, vol. 12, pp. 959–968, 1971.
5. K. Nôno, “Hyperholomorphic functions of a quaternion variable,” Bulletin of Fukuoka University of Education, vol. 32, pp. 21–37, 1982.
6. K. Nôno, “Domains of Hyperholomorphic in ${C}^{2}×{C}^{2}$,” Bulletin of Fukuoka University of Education, vol. 36, pp. 1–9, 1987.
7. J. Kajiwara, X. D. Li, and K. H. Shon, “Function spaces in complex and Clifford analysis, inhomogeneous caychy riemann system of quaternion and Clifford analysis in ellipsoid,” in Proceedings the 14th International Conferences on Finite or Infinite Dimensional Complex Analysis and Applications, vol. 14, pp. 127–155, Hue University, Hue, Vietnam, 2006.
8. S. J. Lim and K. H. Shon, “Properties of hyperholomorphic functions in Clifford analysis,” East Asian Mathematical Journal, vol. 28, pp. 553–559, 2012.
9. S. J. Lim and K. H. Shon, “Hyperholomorphic functions and hyper-conjugate harmonic functions of octonion variables,” Journal of Inequalities and Applications, vol. 77, pp. 1–8, 2013.
10. S. J. Lim and K. H. Shon, “Regularities of functions with values in C(n) of matrix algebras M(n;C),” Journal of Inequalities and Applications. Preprint.
11. B. Kenwright, “A beginners guide to dual-quaternions: what they are, how they work, and how to use them for 3D character hierarchies,” in Proceedings of the 20th International Conferences on Computer Graphics, Visualization and Computer Vision, pp. 1–10, 2012.
12. E. Pennestrì and R. Stefanelli, “Linear algebra and numerical algorithms using dual numbers,” Multibody System Dynamics, vol. 18, no. 3, pp. 323–344, 2007.
13. L. Kula and Y. Yayli, “Dual split quaternions and screw motion in Minkowski 3-space,” Iranian Journal of Science and Technology A, vol. 30, no. 3, pp. 245–258, 2006.
14. Lê Hùng Son, “An extension problem for solutions of partial differential equations in ${ℝ}^{n}$,” Complex Variables, vol. 15, no. 2, pp. 87–92, 1990.
15. L.H. Son, “Extension problem for functions with values in a Clifford algebra,” Archiv der Mathematik, vol. 55, no. 2, pp. 146–150, 1990.
16. L. H. Son, “Extension problem for the solutions of partial differential equations in ${ℝ}^{n}$,” Complex Variables, vol. 18, no. 1-2, pp. 135–139, 1992.
17. R. Rocha-Chávez, M. Shapiro, and F. Sommen, Integral theorems for functions and differential forms in Cm, vol. 428 of Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.
18. L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, Amsterdam, The Netherlands, 1966.