CHARACTERIZATION OF A REGULAR FUNCTION WITH VALUES IN DUAL QUATERNIONS

Abstract

In this paper, we provide the notions of dual quaternions and their algebraic properties based on matrices. From quaternion analysis, we give the concept of a derivative of functions and and obtain a dual quaternion Cauchy-Riemann system that are equivalent. Also, we research properties of a regular function with values in dual quaternions and relations derivative with a regular function in dual quaternions.

1. INTRODUCTION

Let T be the set of quaternion numbers constructed over a real Euclidean quadratic four dimensional vector space. In 2004 and 2006, Kajiwara, Li and Shon [2, 3] obtained some results for the regeneration in complex, quaternion and Clifford analysis, and for the inhomogeneous Cauchy-Riemann system of quaternions and Clifford analysis in ellipsoid. Naser [12] and Nôno [13] obtained some properties of quaternionic hyperholomorphic functions. In 2011, Koriyama, Mae and Nôno [8, 9] researched for hyperholomorphic functions and holomorphic functions in quaternion analysis. Also, they obtained some results of regularities of octonion functions and holomorphic mappings. In 2012, Gotô and Nôno [1] researched for regular functions with values in a commutative subalgebra of matrix algebra . Lim and Shon [10, 11] obtained some properties of hyperholomorphic functions and researched for the hyperholomophic functions and hyperconjugate harmonic functions of octonion variables, and for the dual quaternion functions and its applications. Recently, we [4, 5, 6, 7] obtained some results for the regularity of functions on the ternary quaternion and reduced quaternion field in Clifford analysis, and for the regularity of functions on dual split quaternions in Clifford analysis. Also, we investigated the corresponding Cauchy-Riemann systems in special quaternions and properties of each regular functions defined by the corresponding differential operators in special quaternions.

The aim of the paper is to define the representations of dual quaternions, written by a matrix form. Also, we research the conditions of the derivative of functions with values in dual quaternions and the definition of a regular function for Cauchy-Riemann system in dual quaternions.

 

2. PRELIMINARIES

The field T of quaternions

is a four dimensional non-commutative real field such that its four base elements e0 = 1, e1, e2 and e3 satisfying the following :

The element e0 = 1 is the identity of T . Identifying the element e1 with the imaginary unit in the complex field of complex numbers. The dual numbers extended the real numbers by adjoining one new non-zero element ε with the property ε2 = 0. The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = x + εy with x and y uniquely determined real numbers. Dual numbers form the coefficients of dual quaternions. If we use matrices, dual numbers can be represented as

The sum and product of dual numbers are then calculated with ordinary matrix addition and matrix multiplication; both operations are commutative and associative within the algebra of dual numbers.

 

3. DUAL QUATERNIONS

The algebra

where , is a non-commutative subalgebra of .

We define that the dual quaternionic multiplication of two dual quaternions

and

is given by

The dual quaternionic conjugate Z* of Z is

Then the modulus |Z| and the inverse Z−1 of Z in are defined by the following :

and

By using the multiplication of , the power of Z is for ,

and the division of two can be computed as

Since and are real variables, it can be written by

where and

We use the following differential operators :

where (k = 1, 2) are usual complex differential operations.

The Laplacian operator is

Let S be a bounded open subset in T × T . A function F(Z) is defined by the following form in S with values in :

where uj = uj(x0, x1, x2, x3, y0, y1, y2, y3) and vj = vj(x0, x1, x2, x3, y0, y1, y2, y3) are real valued functions.

Remark 3.1. Using differential operators, we have the following equations:

where

Definition 3.2. Let S be a bounded open subset in T × T . A function F = f + εg is said to be M-regular in S if f and g of F are continuously differential quaternion valued functions in S such that D*F = 0.

Remark 3.3. The equation D*F = 0 is equivalent to

Also, it is equivalent to

The above system is called a dual quaternion Cauchy-Riemann system in dual quaternions.

Let ­ Ω be an open subset of , for Z0 = z0 + εω0 ∈ Ω­,

is called a dual-quaternion function in .

Definition 3.4. A function F is said to be continuous at Z0 = z0 + εω0 if

where the limit has

Definition 3.5. The dual quaternion function F is said to be differentiable in dual quaternions if the limit

exists and the limit is called the derivative of F in dual quaternions.

Remark 3.6. From the definition of derivative of f and properties of differential operations of quaternion valued functions, we have

where is a constant in a domain of f (see [2, 11]). Since the equation (3.2) is equivalent to Dzf, we can express . Hence, by the representations of DF and properties of limit, calculating the division for

Therefore, we can represent

Theorem 3.7. Let F = f + εg be a dual quaternion function in ­. If F satisfies the equation Df = 0, then the derivative of F satisfies the following equation:

Proof. By the division of dual quaternions, we have

Then, the limit

exists if and only if has two cases to deal with

Case 1)

If

then the limit exists and the derivative can be written by

Case 2)

If

then the limit exists and the derivative can be written by

Therefore, the equation is obtained.

Theorem 3.8. Let F = f + εg be a dual quaternion function in ­. If F is a M-regular function in dual quaternions, that is, F satisfies the equation D*F = 0, then the derivative of F satisfies the following equation:

Proof. From the proof of Theorem 3.7, we have

Since F satisfies a dual quaternion Cauchy-Riemann system (3.1), we have

Therefore, since we have