Abstract and Applied Analysis

Volume 2013 (2013), Article ID 136120, 7 pages

http://dx.doi.org/10.1155/2013/136120

Research Article

Regular Functions with Values in Ternary Number System on the Complex Clifford Analysis

Ji Eun Kim, Su Jin Lim, and Kwang Ho Shon

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

Received 21 October 2013; Accepted 5 December 2013

Academic Editor: Junesang Choi

Copyright © 2013 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.

Abstract.

We define a new modified basis which is an association of two bases, and . We give an expression of the form , where is a real number and is a complex number on three-dimensional real skew field. And we research the properties of regular functions with values in ternary field and reduced quaternions by Clifford analysis.

1. Introduction

The noncommutative three-dimensional real field of the hypercomplex numbers is called a ternary number system . The quaternions are represented by the form , where () are real numbers on four dimensional real field . Fueter [1] has given a definition of quaternionic functions in and Deavours [2] and Sudbery [3] have developed theories of quaternionic analysis. Naser [4] investigated some properties of hyperholomorphic functions and Koriyama et al. [5] researched properties of hyperholomorphic functions and holomorphic functions in quaternionic analysis. Nôno [6] obtained several results for regular functions which have a complex number form in quaternion analysis. Cho [7] researched some properties of Euler’s formula and De moivre’s formula for quaternions. Sangwine and Bihan [8] obtained some results for the quaternionic polar representation with a complex modulus and complex argument inspired by the cayley-dickson form. Fueter [9] obtained some properties of the three variables which are called the Fueter variables and researched the fact that structures lead to the set of all Fueter-regular functions in the general cases of Clifford analysis. By Brackx et al. [10], the theory of Fueter-regularity has been developed and generalized as quaternionic variables for theories of Clifford-valued regular functions.

Lim and Shon [11–13] researched the existence of hyperconjugate harmonic functions of octonion variables, properties of dual quaternion functions, and regularity of functions with values in a noncommutative subalgebra of complex matrix algebras.

We consider that ternary numbers are generated by a new basis and give some properties of regular functions with values in . Also, we represent the corresponding Euler’s formula for the form and investigate calculating rules for regular functions in Clifford analysis. We research new representations of Fueter variables in reduced quaternions with and some characteristics of regularity of functions on the Fueter variable system.

2. Preliminaries

The ternary number system is a three dimensional non-commutative and associative real field by three bases , , and with the following rules:

The element is the identity of and identifies the imaginary unit in the complex field. We consider an association of two bases and as follows:

where , , and , are real numbers except both zeros.

The number of the skew field is

where () are real variables, , and .

We define the ternary number system

The conjugate number of in is given by the form:

And the norm of and the inverse of are given by the following forms:

where and .

We define the addition and multiplication of two ternary numbers and as follows:

Theorem 1. Let be an arbitrary number in . Then the corresponding Euler formula for is

Moreover, taking logarithms of both sides, one obtains the equation as follows:

Proof. For the number in , we get and . Then,

From

we have

We consider the following differential operators:

where and . Then the Laplacian operator is

Let be an open set in . The function that is defined by the following form in with values in :

satisfies

where () are real-valued functions and

are complex-valued functions with values in .

Remark 2. The operators and act for the function on as follows:

3. Properties of Regular Functions with Values in

Definition 3. Let be an open set in . A function is said to be L(R)-regular in , if the following two conditions are satisfied:

(i) and are continuously differential functions on ;(ii) () on .

Remark 4. The left equation (ii) of Definition 3 is equivalent to the following:

The equations in (19) are called the corresponding Cauchy-Riemann system for in . The right equation (ii) of Definition 3 is equivalent to (19). When the function is a L-regular function on , simply we say that is a regular function on . In this case, we often say that is a biregular function on .

Remark 5. Let be an open set in and let be a regular function on . Then we can replace the corresponding Cauchy-Riemann system in as follows:

where () are real-valued functions.

Theorem 6. Let be an open set in and let be a regular function on . Then the derivative of defined by is

on .

Proof. By the definition of regular function with values in , we have

on . And

on .

Theorem 7. Let be an open set in and let be a function with values in . Suppose that and exist and are continuous on . If

on , then is regular on .

Proof. Since , we have

Hence, we have and then is regular on .

Definition 8. Let be an open set in . A function is said to be harmonic on if all its components and of are harmonic on .

Proposition 9. Let be an open set in . If the function is regular on , then is harmonic on .

Proof. Since is regular function on , we have

Similarly, we can prove that . So, we obtain the result.

Proposition 10. Let be an open set in and let and be regular functions on . Then the following properties hold:

(i) is regular on , if is any ternary constant;(ii) is not regular on , if is any ternary constant;(iii) is regular on ;(iv) is not regular on . Moreover, if is a real-valued function, then is regular on .

Proof. It is sufficient to show the second condition of Definition 3.

(i)Let be a ternary constant with , where

and , , , , and are real numbers. Then the equation

Hence, is regular on .(ii)Since

is not zero, is not always regular on .(iii)Since

is regular on .(iv)Since

is not zero, is not always regular on .

Theorem 11. Let be an open set in and let and be regular functions on . Then we have the following equations:

Proof. From the proof of Proposition 10, we have the following equations:

Similarly, we can prove (33).

We let

Theorem 12. Let be an open set in and be any domain in with smooth boundary such that . If is a regular function on , then

where is the ternary product of the form on the function .

Proof. Since the function exists, we have

Then

where in , and by the corresponding Cauchy-Riemann system for in , . By Stokes theorem, we obtain the result.

Remark 13. Since

we have

where

And is the greatest integer that is less than or equal to .

Theorem 14. Let be a homogeneous polynomial of degree with respect to the variables and . If is regular on , then

where is a nonnegative integer.

Proof. Since is a homogeneous polynomial, then

Also, since is a homogeneous polynomial of degree , we have

Then we have

Continuing this process, we can get the result (42). Similarly, we obtain the result (43).

4. Properties of Regular Functions with Values in

We define the number system

where and .

The non-commutative multiplication of two numbers and is defined by

The conjugate number of in is given by the following:

And the norm of and the inverse of are given by the following forms:

We consider the following differential operators:

where

Then the Laplacian operator is

Let be an open set in . The function that is defined by the following form in with values in :

satisfies

where and are complex-valued functions with values in and () are real-valued functions.

Remark 15. The operators and act for a function on as follows:

We define a commutative multiplication of two numbers and by

Remark 16. The operators and act for a function on as follows:

Definition 17. Let be a domain in . A function is said to be dot-regular in if the following two conditions are satisfied:

(i) and are differential functions in ,(ii) in .

Remark 18. The above equation (ii) of Definition 17 is equivalent as follows:

Theorem 19. Let be an open set in and let be a dot-regular function on . Then the derivative of defined by is

Proof. By the definition of a dot-regular function with values in , we have

on . And, similarly, we have

on .

Acknowledgment

Kwang Ho Shon was supported by the Research Fund Program of Research Institute for Basic Sciences, Pusan National University, Korea, 2013, Project no. RIBS-PNU-2013-101.

References

R. Fueter, “Die funktionentheorie der differentialgleichungen und mit vier reellen variablen,” Commentarii Mathematici Helvetici, vol. 7, no. 1, pp. 307–330, 1935.

C. A. Deavours, “The quaternion calculus,” The American Mathematical Monthly, vol. 80, pp. 995–1008, 1973.

A. Sudbery, “Quaternionic analysis,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 85, no. 2, pp. 199–224, 1979.

M. Naser, “Hyperholomorphic functions,” Silberian Mathematical Journal, vol. 12, pp. 959–968, 1971.

H. Koriyama, H. Mae, and K. Nôno, “Hyperholomorphic functions and holomorphic functions in quaternionic analysis,” Bulletin of Fukuoka University of Education, vol. 60, pp. 1–9, 2011.

K. Nôno, “Hyperholomorphic functions of a quaternion variable,” Bulletin of Fukuoka University of Education, vol. 32, pp. 21–37, 1983.

E. Cho, “De moivre's formula for quaternions,” Applied Mathematics Letters, vol. 11, no. 6, pp. 33–35, 1998.

S. J. Sangwine and N. L. Bihan, “Quaternion polar representation with a complex modulus and complex argument inspired by the cayley-dickson form,” Advances in Applied Clifford Algebras, vol. 20, no. 1, pp. 111–120, 2010.

R. Fueter, “Die theorie der regulären funktionen einer quaternionenvariablen,” in Comptès Rendus du Congrès International des Mathèmaticiens, Oslo 1936, vol. 1, pp. 75–91, 1935.

F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis, vol. 76 of Research Notes in Mathematics, 1982.

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.

S. J. Lim and K. H. Shon, “Regularity of functions with values in a non-commutative algebra of complex matrix algebras,” Preprint