Research Article | Open Access

# Regularity of Functions on the Reduced Quaternion Field in Clifford Analysis

**Academic Editor:**Junesang Choi

#### Abstract

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 .

#### 1. Introduction

Meglihzon [1], Sudbery [2], and Fueter [3] 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 [2], Soucek [4], and Sommen [5] attempted to research the Cauchy approach using differential forms and differential operators in Clifford analysis. Fueter [3] and Naser [6] 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 [2] 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 [12] 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 [13] and Koriyama et al. [14] 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.

#### 2. Preliminaries

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
**
where .**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 [19],
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

*Remark 2. *By Theorem 1 and the properties of the Euler formula, if , then we can write
also, if , then
where and are the conjugate Fueter variables of (see [20]).

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 .

In particular, the equation of Definition 4 is equivalent to Moreover, (38) is equivalent to the following system: The above system is a corresponding Cauchy-Riemann system in .

*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 .*

* Proof. *Since , we have
where in . From the corresponding Cauchy-Riemann system (39) for in , we have the system (56). Hence, and, therefore, by Stokes theorem, we obtain the following result:

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The third author was supported by a 2-Year Research Grant of Pusan National University.

#### References

- A. S. Meglihzon, “Po povodu monogennosti kvaternionov,”
*Do-Klady Akademii Nauk SSSR 3*, vol. 59, pp. 431–434, 1948. View at: Google Scholar - A. Sudbery, “Quaternionic analysis,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 85, no. 2, pp. 199–224, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Fueter, “Die Funktionentheorie der Differentialgleichungen $\mathrm{\Delta}u=0$ und $\mathrm{\Delta}\mathrm{\Delta}u=0$ mit vier reellen Variablen,”
*Commentarii Mathematici Helvetici*, vol. 7, no. 1, pp. 307–330, 1934. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Naser, “Hyperholomorphic functions,”
*Siberian Mathematical Journal*, vol. 12, pp. 959–968, 1971. View at: Google Scholar - K. Nôno, “Hyperholomorphic functions of a quaternion variable,”
*Bulletin of Fukuoka University of Education III*, vol. 32, p. 21, 1983. View at: Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Google Scholar | Zentralblatt MATH | MathSciNet - K. Nōno, “Domains of hyperholomorphy in ${C}^{2}\times {\u2102}^{2}$,”
*Bulletin of Fukuoka University of Education III*, vol. 36, pp. 1–9, 1987. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. Ryan, “Complexified Clifford analysis,”
*Complex Variables and Elliptic Equations*, vol. 1, no. 1, pp. 119–149, 1982/83. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Ryan, “Special functions and relations within complex Clifford analysis. I,”
*Complex Variables and Elliptic Equations*, vol. 2, no. 2, pp. 177–198, 1983. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Malonek, “A new hypercomplex structure of the Euclidean space ${\mathbb{R}}^{m+1}$ and the concept of hypercomplex differentiability,”
*Complex Variables: Theory and Applications*, vol. 14, no. 1–4, pp. 25–33, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Gotô and K. Nôno, “Regular functions with values in a commutative subalgebra $\u2102(\u2102)$ of matrix algebra $M(4,\mathbb{R})$,”
*Bulletin of Fukuoka University of Education III*, vol. 61, pp. 9–15, 2012. View at: Google Scholar | MathSciNet - 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. View at: Google Scholar | MathSciNet - 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. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - 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. View at: Google Scholar - 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. View at: Google Scholar - 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. View at: Publisher Site | Google Scholar | MathSciNet - D. H. Titterton and J. L. Weston, “Strapdown inertial navigation technology,”
*Peter Pregrinus*, 1997. View at: Google Scholar - 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. View at: Google Scholar

#### Copyright

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.