Regularity of Functions on the Reduced Quaternion Field in Clifford Analysis

and Applied Analysis 3 Similarly, if x 2 = tπ (t ∈ Z), then exp (z) = exp (x 0 ) {(−1) t (cos (x 1 ) + e 1 sin (x 1 ))} = (−1) t exp (x 0 ) exp (e 1 x 1 ) . (17) Further, by the Euler formula and the addition rule of trigonometric functions, exp (z) = exp (e 1 z 1 + e 2 z 2 ) = exp (e 1 z 1 ) exp (e 2 z 2 ) = (cos (z 1 ) + e 1 sin (z 1 )) (cos (z 2 ) + e 2 sin (z 2 )) = {cos (x 1 ) cos(e 1 x 0 2 ) + sin (x 1 ) sin(e 1 x 0


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][8][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 C, R, and Z, respectively, the field of complex numbers, the field of real numbers, and the set of all integers.We [15,16] showed that any complexvalued harmonic function  1 in a pseudoconvex domain  of C 2 × C 2 has a hyperconjugate harmonic function  2 in  such that the quaternion-valued function  1 +  2  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 R 3 and investigate a hypercomplex structure of R 3 .We define regular functions and obtain the representation of the corresponding Cauchy-Riemann equations for regular functions in the reduced quaternion field.

Preliminaries
A three-dimensional, noncommutative, and associative real field, called a ternary number system, is constructed by three base elements  0 ,  1 , and  2 which satisfy In addition, let  0 be the identity of a ternary number system and  1 identifies the imaginary unit √ −1 in the complex field, and where   =   − (1/2)   0 ( = 1, 2) and   ( = 0, 1, 2) are real variables.They satisfy the equations where , and   ( = 0, 1, 2) are real variables.
For any two elements  =  1  1 + 2  2 and  =  1  1 + 2  2 of C(T), 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  −1 of  in C(T) are given by For any element  in C(T), we have the corresponding exponential function   denoted by Theorem 1.Let  be an arbitrary number in C(T).Then the corresponding exponential function exp() of  in C(T) is given as where ,  ∈ Z. Furthermore, as hyperbolic functions, one has where ,  ∈ Z.
Let Ω be an open subset of R 3 and let a function () be defined by the following form on Ω with values in C(T): satisfying where and   ( = 0, 1, 2) are real-valued functions.
From the chain rule, we use the following differential operators: where in C(T).We have the following equations: and then, the operator / operates to  as follows: Thus, we have a corresponding Laplacian in the reduced quaternion C(T): From the definition of the differential operators in C(T), we have and, therefore, Similarly, we have Moreover, (38) is equivalent to the following system: (39) The above system is a corresponding Cauchy-Riemann system in C(T).(40) Also, the above equation ( 40) is equivalent to the following system: ).
(41) Further, the above system (41) is also a corresponding Cauchy-Riemann system in C(T).Since the system (39) is equivalent to the system (41), we say that () of Definition 4 is a regular function on Ω ⊂ R 3 .When the function () is either an L-regular function or an R-regular function on Ω ⊂ R 3 , we simply say that () is a regular function on Ω ⊂ R 3 .

Properties of Regular Functions with Values in C(T)
We define the derivative   () of () by the following: Proposition 6.Let Ω be an open set in R 3 and let a function () be a regular function defined on Ω.Then Proof.From the definition of a regular function (: (/ * ) = 0), we have Therefore,