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The Regularity of Functions on Dual Split Quaternions in Clifford Analysis
This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions and provides differential operators in dual split quaternions and a dual split regular function on that has a dual split Cauchy-Riemann system in dual split quaternions.
Hamilton introduced quaternions, extending complex numbers to higher spatial dimensions in differential geometry (see ). A set of quaternions can be represented as where , , and denotes the set of real numbers. Cockle  introduced a set of split quaternions as where , , and . A set of split quaternions is noncommutative and contains zero divisors, nilpotent elements, and nontrivial idempotents (see [3, 4]). Previous studies have examined the geometric and physical applications of split quaternions, which are required in solving split quaternionic equations (see [5, 6]). Inoguchi  reformulated the Gauss-Codazzi equations in forms consistent with the theory of integrable systems in the Minkowski 3-space for split quaternion numbers.
A dual quaternion can be represented in a form reflecting an ordinary quaternion and a dual symbol. Because dual-quaternion algebra is constructed from real eight-dimensional vector spaces and an ordered pair of quaternions, dual quaternions are used in computer vision applications. Kenwright  provided the characteristics of dual quaternions, and Pennestrì and Stefanelli  examined some properties by using dual quaternions. Son [10, 11] offered an extension problem for solutions of partial differential equations and generalized solutions for the Riesz system. By using properties of Hamilton operators, Kula and Yayli  defined dual split quaternions and gave some properties of the screw motion in the Minkowski 3-space, showing that has a rotation with unit split quaternions in and a scalar product that allows it to be identified with the semi-Euclidean space for split quaternion numbers.
It was shown (see [12, 13]) that any complex-valued harmonic function in a pseudoconvex domain of , being the set of complex numbers, has a conjugate function in such that the quaternion-valued function is hyperholomorphic in and gave a regeneration theorem in a quaternion analysis in view of complex and Clifford analysis. In addition, we [14, 15] provided a new expression of the quaternionic basis and a regular function on reduced quaternions by associating hypercomplex numbers and . We  investigated the existence of hyperconjugate harmonic functions of an octonion number system, and we [17, 18] obtained some regular functions with values in dual quaternions and researched an extension problem for properties of regular functions with values in dual quaternions and some applications for such problems.
This paper provides a regular function and some properties of differential operators in dual split quaternions. In addition, we research some equivalent conditions for Cauchy-Riemann systems and expressions of power series in dual split quaternions from the definition of dual split regular on an open set .
A dual number has the form , where and are real numbers and is a dual symbol subject to the rules and a split quaternion is an expression of the form where and are split quaternionic units satisfying noncommutative multiplication rules (for split quaternions, see ): Similarly, a dual split quaternion can be written as which has elements of the following form: where and are split quaternion components, , , , and are usual complex numbers, and . The multiplication of split quaternionic units with a dual symbol is commutative . However, by properties of split quaternionic unit, where with , , , and . For instance, Because of the properties of the eight-unit equality, the addition and subtraction of dual split quaternions are governed by the rules of ordinary algebra. Here the symbol is used by just enumerating and , not times . For example, and .
For any two elements and of , where and are split quaternion components and , their noncommutative product is given by The conjugation of and the corresponding modulus in are defined by where and .
Lemma 1. For all and , we have
Let be an open subset of . Then the function can be expressed as where the component functions are split quaternionic-valued functions. The component functions are where and are complex-valued functions, and and are real-valued functions.
Now, we let differential operators and be defined on as Then the conjugate operators and are where act on . These operators are called corresponding Cauchy-Riemann operators in , where and are usual differential operators used in the complex analysis.
Remark 2. From the definition of differential operators on , where .
Definition 3. Let be an open set in . A function is called an (resp., )-regular function on if the following two conditions are satisfied:(i) are continuously differential functions on , and(ii) (resp., ) on .
In particular, the equation of Definition 3 is equivalent to In addition, Concretely, the following system is obtained: The above systems (23) and (24) are corresponding Cauchy-Riemann systems in . Similarly, the equation of Definition 3 is equivalent to Then, Concretely, the following system is obtained: The above systems (26) and (27) are corresponding Cauchy-Riemann systems in .
On the other hand, the equation of Definition 3 is equivalent to Then, where Concretely, the following system is obtained: Similarly, the equation of Definition 3 is equivalent to Then, where Concretely, the system is obtained as follows: From the systems (24), (27), (31), and (35), the equations and are different. Therefore, the equations and should be distinguished as -regular functions and -regular functions on , respectively. Now the properties of the -regular function with values in are considered.
3. Properties of -Regular Functions with Values in
We consider properties of a -regular functions with values in .
Theorem 4. Let be an open set in and let be an -regular function defined on . Then
Proof. By the system (23), we have Therefore, we obtain
Theorem 5. Let be an open set in and be an -regular function defined on . Then
Proof. By the system (26), we have Therefore, we obtain the following equation:
Proposition 6. From properties of differential operators, the following equations are obtained:
Proof. By properties of the power of dual split quaternions and derivatives on , the following derivatives are obtained: The other equations are calculated using a similar method, and the above equations are obtained.
Theorem 7. Let be an open set in and let be a function on with values in . Then the power of in is not an -regular function but an -regular function on , where .
Proof. From the definition of the -regular function on and Proposition 6, we may consider whether the power of in satisfies the equation . Since , Hence, the power of is not -regular on . On the other hand, from the equations in Proposition 6, we have , , and . Then, Therefore, by the definition of the -regular function on , a power of is -regular on .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and Future Planning (2013R1A1A2008978).
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Copyright © 2014 Ji Eun Kim 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.