TAYLOR SERIES OF FUNCTIONS WITH VALUES IN DUAL QUATERNION

Abstract

We define an ${\varepsilon}$-regular function in dual quaternions. From the properties of ${\varepsilon}$-regular functions, we represent the Taylor series of ${\varepsilon}$-regular functions with values in dual quaternions.

1. INTRODUCTION

Fueter [2] and Naser [8] have studied properties of quaternionic differential equations as a generalization of the extended Cauchy-Riemann equations in the complex holomorphic function theory and Nôno [9, 10, 11] has given a definition of regular functions over the quaternion field identified with In 1979, Sudbery [15] developed quaternionic regular function theories. By using a generalization of the Cauchy-Riemann equation, Ryan [12, 13] has developed regular function theories on complex Cliffrd algebra of quaternion valued functions.

In 1873, Clifford [1] originally conceived the algebra of dual numbers. Dual algebra has been often used for closed form solutions in the field of displacement analysis. Kotelnikov [6] and Study [14] developed dual vectors and dual quaternions for use in the application of mechanics and realized that this associative algebra was ideal for describing the group of motions of three-dimensional spaces.

In 2011, Koriyama, Mae and Nôno [5] investigated hyperholomorphic functions and holomorphic functions in quaternion analysis. In 2012, Gotô and Nôno [3] researched regular functions with values in a commutative subalgebra of matrix algebra in four real dimension and we [7] obtained regularities of functions with values in subalgebra of matrix algebras in complex n-dimensional.

In this paper, we introduce the dual quaternion numbers and give some properties of ε-regular functions in dual quaternions by using the associated Pauli matrices. We give the notation of the derivative for functions with values in dual quaternions and obtain the representation of the Taylor series of ε-regular functions.

 

2. PRELIMINARY

A dual quaternion is an ordered pair of quaternions and is constructed from eight base elements e0, e1, e2, e3, ε, e1ε, e2ε and e3ε. We consider the associated Pauli matrices where . And, we let the dual quaternion identity which is a nonzero and satisfy 0ε = ε0 = 0, 1ε = ε1 = ε, ε2 = 0 and where , and is a dual quaternion component of The element e0 is the identity, the element ε is the dual identity of and the element e1 identifies the imaginary unit in the ℂ-field of complex numbers. We can identify with

The dual quaternionic conjugation z∗ of z, the absolute value |z| of z and an inverse z−1 of z in are defined, respectively, by where and .

Let Ω be an open subset of and the dual quaternion function f : Ω → satisfy where and are real-valued functions.

We use the following two dual quaternion differential operators which are defined as and where Then we have where

Definition 2.1. Let Ω be an open set in . A function f(z) is said to be ε-regular in Ω if the following two conditions are satisfied:

(a) fj (j = 0,1,2,3) are continuously diffrential functions in Ω, and

(b) D∗f(z) = 0 in Ω.

 

3. TAYLOR SERIES OF DUAL QUATERNION FUNCTIONS

We define the derivative f'(z) of f(z) by the following: f'(z) := Df(z)

Lemma 3.1. Let Ω be a domain in and f(z) be a holomorphic mapping and ε-regular defined in Ω. Then, Proof. Since f(z) is an ε-regular function in Ω, we have where From we can get

Theorem 3.2. Let f(z) be a homogeneous polynomial of degree m with respect to the variables ξ and ξ?. If f(z) is a holomorphic and ε-regular function in then we have

Proof. Since f(z) is a homogeneous polynomial, we have

Then f'(z) = mξm-1+εm(m - 1)ξm-2ξ*f'(z)z = mξm+εm2ξm-1ξ*.

Thus, . And f''(z) = m(m - 1)ξm-2+εm(m - 1)(m - 2)ξm-3ξ*f''(z)z = m(m - 1)ξm-1+εm(m - 1)2ξm-2ξ*.

Thus, Repeating the above calculation, we have

Theorem 3.3. Let Ω be a domain in Let f(z) be a holomorphic and ε-regular function in Ω and α ∈ Ω. Then there exists a neighborhood Uα of α such that where .

Proof. From substituting a dual number into Taylor series, we have

By Theorem 3.2 and zm = ξm + ε((m − 1)ξm−1ξ∗ + ξm−2ξ∗), we have

Remark 3.4. Let Ω be a domain in . If g0(z) is a holomorphic function with value in quaternions, then there exists a function g1(z) with value in quaternions such that f(z) = g0(z)+εg1(z) is ε-regular in Ω.

By the results of Kenwright [4], f(z) = f(ξ)+εf′(ξ)ξ∗.

We put g0(z) = f(ξ), then Dg0(z) = f′(ξ). Hence we put g1(z) = Dg0(z)ξ∗, we have f(z) = g0(z)+εg1(z).

Example 3.5. Let Ω be a domain in If g0 = sin(nζ), then g′0 = n cos(nξ), n∈ℤ. Thus, there exists a function g1 = n cos(nξ)ξ* such that f(z) = g0(z)+εg1(z). is ε-regular in Ω.