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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 654798, 8 pages
Regularity of Functions on the Reduced Quaternion Field in Clifford Analysis
Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
Received 11 December 2013; Revised 12 February 2014; Accepted 18 February 2014; Published 20 March 2014
Academic Editor: Junesang Choi
Copyright © 2014 Ji Eun Kim et al. 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.
We define a new hypercomplex structure of and a regular function with values in that structure. From the properties of regular functions, we research the exponential function on the reduced quaternion field and represent the corresponding Cauchy-Riemann equations in hypercomplex structures of .
Meglihzon , Sudbery , and Fueter  demonstrated that there are three possible approaches (the Cauchy approach, Weierstrass approach, and Riemann approach) in the theories of functions that would generalize holomorphic functions with respect to several complex variables. Sudbery , Soucek , and Sommen  attempted to research the Cauchy approach using differential forms and differential operators in Clifford analysis. Fueter  and Naser  studied the properties of quaternionic differential equations as a generalization of the extended Cauchy-Riemann equations in the complex holomorphic function theory. Nôno [7–9] and Sudbery  gave a definition and the development of regular functions over the quaternion field. Ryan [10, 11] developed the theories of regular functions in a complex Clifford analysis using a generalization of the Cauchy-Riemann equation. Malonek  considered analogously the function theory of hypercomplex variables. He defined the hypercomplex differentiability for the existence of a function over the Clifford algebra and monogenicity based on a generalized Cauchy-Riemann system. Gotô and Nôno  and Koriyama et al.  dealt with differential operators with the derivative of regular functions in quaternion.
We shall denote by , , and , respectively, the field of complex numbers, the field of real numbers, and the set of all integers. We [15, 16] showed that any complex-valued harmonic function in a pseudoconvex domain of has a hyperconjugate harmonic function in such that the quaternion-valued function is hyperholomorphic in and gave a regeneration theorem in quaternion analysis in the view of complex and Clifford analysis. Further, we [17, 18] investigated the existence of the hyperconjugate harmonic functions of the octonion number system and some properties of dual quaternion functions.
In this paper, we introduce the Fueter variables on and investigate a hypercomplex structure of . We define regular functions and obtain the representation of the corresponding Cauchy-Riemann equations for regular functions in the reduced quaternion field.
A three-dimensional, noncommutative, and associative real field, called a ternary number system, is constructed by three base elements , , and which satisfy In addition, let be the identity of a ternary number system and identifies the imaginary unit in the complex field, and where and are real variables. They satisfy the equations where , , , and are real variables.
For any two elements and of , their product is given by where the corresponding commutative inner product satisfies and the corresponding noncommutative outer product satisfies The conjugation , the corresponding norm , and the inverse of in are given by
For any element in , we have the corresponding exponential function denoted by
Theorem 1. Let be an arbitrary number in . Then the corresponding exponential function of in is given as
Furthermore, as hyperbolic functions, one has where .
Proof. For any element of , Since a scalar part of is , a vector part of is , and , by , and, similarly, we have Then we have Also, we obtain Since (15) has to be equal to (14), , that is, or . Therefore, or , and then or , where . If , then Similarly, if , then Further, by the Euler formula and the addition rule of trigonometric functions, Since and , we have Since we obtain and, similarly, Since (22) has to be equal to (21), , that is, or . Therefore, or , and then or , where . If , then Similarly, if , then
Let be an open subset of and let a function be defined by the following form on with values in : satisfying where , and are real-valued functions.
From the chain rule, we use the following differential operators: where in . We have the following equations: and then, the operator operates to as follows: Thus, we have a corresponding Laplacian in the reduced quaternion :
Remark 3. Let Ω be an open set of . From the definition of the differential operators in , we have and, therefore, Similarly, we have and, therefore,
Definition 4. Let be an open set in and for any element in . A function is said to be L(R)-regular on if the following conditions are satisfied:(i) are continuously differential functions on , and(ii) on .
Remark 5. From the multiplications of , the equation of Definition 4 is equivalent to Also, the above equation (40) is equivalent to the following system: Further, the above system (41) is also a corresponding Cauchy-Riemann system in . Since the system (39) is equivalent to the system (41), we say that of Definition 4 is a regular function on . When the function is either an L-regular function or an R-regular function on , we simply say that is a regular function on .
3. Properties of Regular Functions with Values in
We define the derivative of by the following:
Proposition 6. Let be an open set in and let a function be a regular function defined on . Then
Proof. From the definition of a regular function , we have Therefore, Hence, we obtain the equation Similarly, by calculating the derivative of , Therefore, we have the equation Further, using the same procedure, we obtain the equations
Proposition 7. Let be an open set in . If is a regular function on , then we have where is a positive integer.
Proof. Since is a regular function on with values in , by Definition 4, Hence, is a regular function with values in . From Proposition 6, we have By repeating the above process, we can obtain the equation
We let on an open set in .
Theorem 8. Let be an open set in . If is a regular function on , then the following equation holds true:
Proof. Since is a regular function on , we have the following system: By the definition of , we have From Proposition 7, we have . Hence, by calculating and comparing the above polynomials, we obtain that is equal to .
Next, we consider a differential form
Theorem 9. Let be an open set in and let be any domain on with a smooth distinguished boundary such that . If is a regular function on , then one has where is the reduced quaternionic product of the form on the function .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The third author was supported by a 2-Year Research Grant of Pusan National University.
- A. S. Meglihzon, “Po povodu monogennosti kvaternionov,” Do-Klady Akademii Nauk SSSR 3, vol. 59, pp. 431–434, 1948.
- A. Sudbery, “Quaternionic analysis,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 85, no. 2, pp. 199–224, 1979.
- R. Fueter, “Die Funktionentheorie der Differentialgleichungen und mit vier reellen Variablen,” Commentarii Mathematici Helvetici, vol. 7, no. 1, pp. 307–330, 1934.
- V. Soucek, Regularni funkce quaternionove promenne [Thesis], Charles University Prague, 1980.
- F. Sommen, “Monogenic differential forms and homology theory,” Proceedings of the Royal Irish Academy A, vol. 84, no. 2, pp. 87–109, 1984.
- M. Naser, “Hyperholomorphic functions,” Siberian Mathematical Journal, vol. 12, pp. 959–968, 1971.
- K. Nôno, “Hyperholomorphic functions of a quaternion variable,” Bulletin of Fukuoka University of Education III, vol. 32, p. 21, 1983.
- K. Nôno, “Characterization of domains of holomorphy by the existence of hyper-conjugate harmonic functions,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 31, no. 2, pp. 159–161, 1986.
- K. Nōno, “Domains of hyperholomorphy in ,” Bulletin of Fukuoka University of Education III, vol. 36, pp. 1–9, 1987.
- J. Ryan, “Complexified Clifford analysis,” Complex Variables and Elliptic Equations, vol. 1, no. 1, pp. 119–149, 1982/83.
- J. Ryan, “Special functions and relations within complex Clifford analysis. I,” Complex Variables and Elliptic Equations, vol. 2, no. 2, pp. 177–198, 1983.
- H. Malonek, “A new hypercomplex structure of the Euclidean space and the concept of hypercomplex differentiability,” Complex Variables: Theory and Applications, vol. 14, no. 1–4, pp. 25–33, 1990.
- S. Gotô and K. Nôno, “Regular functions with values in a commutative subalgebra of matrix algebra ,” Bulletin of Fukuoka University of Education III, vol. 61, pp. 9–15, 2012.
- H. Koriyama, H. Mae, and K. Nôno, “Hyperholomorphic functions and holomorphic functions in quaternionic analysis,” Bulletin of Fukuoka University of Education III, vol. 60, pp. 1–9, 2011.
- J. Kajiwara, X. D. Li, and K. H. Shon, “Regeneration in complex, quaternion and Clifford analysis,” in Finite or Infinite Dimensional Complex Analysis and its Applications, vol. 2 of Advances in Complex Analysis and Its Applications, pp. 287–298, Kluwer Academic, Hanoi, Vietnam, 2004.
- J. Kajiwara, X. D. Li, and K. H. Shon, “Function spaces in complex and Clifford analysis,” in Inhomogeneous Cauchy Riemann System of Quaternion and Clifford Analysis in Ellipsoid, International Colloquium on Finite or Infinite Dimensional Complex Analysis and Its Applications, vol. 14, pp. 127–155, Hue University, Hue, Vietnam, 2006.
- S. J. Lim and K. H. Shon, “Hyperholomorphic fucntions and hyperconjugate harmonic functions of octonion variables,” Journal of Inequalities and Applications, vol. 77, pp. 1–8, 2013.
- S. J. Lim and K. H. Shon, “Dual quaternion functions and its applications,” Journal of Applied Mathematics, vol. 2013, Article ID 583813, 6 pages, 2013.
- D. H. Titterton and J. L. Weston, “Strapdown inertial navigation technology,” Peter Pregrinus, 1997.
- R. Fueter, “Die theorie der regularen funktionen einer quaternionenvariablen,” in Comptés Rendus du Congrès International des Mathenaticiens, vol. 1, pp. 75–91, Oslo, Norway, 1936.