The Regularity of Functions on Dual Split Quaternions in Clifford Analysis

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 369430, 8 pages http://dx.doi.org/10.1155/2014/369430 Research Article The Regularity of Functions on Dual Split Quaternions in Clifford Analysis Ji Eun Kim and Kwang Ho Shon Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea Correspondence should be addressed to Kwang Ho Shon; Received 28 January 2014; Accepted 2 April 2014; Published 17 April 2014 Academic Editor: Zong-Xuan Chen Copyright © 2014 J. E. Kim and K. H. Shon. 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. This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions and provides differential operators in dual split quaternions and a dual split regular function on Ω ⊂ C2 × C2 that has a dual split Cauchy-Riemann system in dual split quaternions. 1. Introduction Hamilton introduced quaternions, extending complex numbers to higher spatial dimensions in differential geometry (see [1]). A set of quaternions can be represented as H = {𝑧 = 𝑥0 + 𝑥1 𝑖 + 𝑥2 𝑗 + 𝑥3 𝑘 : 𝑥𝑚 ∈ R, 𝑚 = 0, 1, 2, 3} , (1) where 𝑖2 = 𝑗2 = 𝑘2 = −1, 𝑖𝑗𝑘 = −1, and R denotes the set of real numbers. Cockle [2] introduced a set of split quaternions as S = {𝑧 = 𝑥0 + 𝑥1 𝑒1 + 𝑥2 𝑒2 + 𝑥3 𝑒3 : 𝑥𝑚 ∈ R, 𝑚 = 0, 1, 2, 3} , (2) where 𝑒12 = −1, 𝑒22 = 𝑒32 = 1, and 𝑒1 𝑒2 𝑒3 = 1. A set of split quaternions is noncommutative and contains zero divisors, nilpotent elements, and nontrivial idempotents (see [3, 4]). Previous studies have examined the geometric and physical applications of split quaternions, which are required in solving split quaternionic equations (see [5, 6]). Inoguchi [7] reformulated the Gauss-Codazzi equations in forms consistent with the theory of integrable systems in the Minkowski 3-space for split quaternion numbers. A dual quaternion can be represented in a form reflecting an ordinary quaternion and a dual symbol. Because dual-quaternion algebra is constructed from real eightdimensional vector spaces and an ordered pair of quaternions, dual quaternions are used in computer vision applications. Kenwright [8] provided the characteristics of dual quaternions, and Pennestrı̀ and Stefanelli [9] examined some properties by using dual quaternions. Son [10, 11] offered an extension problem for solutions of partial differential equations and generalized solutions for the Riesz system. By using properties of Hamilton operators, Kula and Yayli [4] defined dual split quaternions and gave some properties of the screw motion in the Minkowski 3-space, showing that H has a rotation with unit split quaternions in H and a scalar product that allows it to be identified with the semi-Euclidean space for split quaternion numbers. It was shown (see [12, 13]) that any complex-valued harmonic function 𝑓1 in a pseudoconvex domain 𝐷 of C2 × C2 , C being the set of complex numbers, has a conjugate function 𝑓2 in 𝐷 such that the quaternion-valued function 𝑓1 + 𝑓2 𝑗 is hyperholomorphic in 𝐷 and gave a regeneration theorem in a quaternion analysis in view of complex and Clifford analysis. In addition, we [14, 15] provided a new expression of the quaternionic basis and a regular function on reduced quaternions by associating hypercomplex numbers 𝑒1 and 𝑒2 . We [16] investigated the existence of hyperconjugate harmonic functions of an octonion number system, and we [17, 18] obtained some regular functions with values in dual quaternions and researched an extension problem for properties 2 Abstract and Applied Analysis of regular functions with values in dual quaternions and some applications for such problems. This paper provides a regular function and some properties of differential operators in dual split quaternions. In addition, we research some equivalent conditions for CauchyRiemann systems and expressions of power series in dual split quaternions from the definition of dual split regular on an open set Ω ⊂ C2 × C2 . A dual number 𝐴 has the form 𝑎 + 𝜀𝑏, where 𝑎 and 𝑏 are real numbers and 𝜀 is a dual symbol subject to the rules 0𝜀 = 𝜀0 = 0, 1𝜀 = 𝜀1 = 𝜀, 2 𝜀 = 0, (3) 𝑝(02) = 𝑧0 + 𝑧1 𝑒2 = 𝑥0 − 𝑥1 𝑒1 + 𝑥2 𝑒2 − 𝑥3 𝑒3 , 𝑝(03) = 𝑧0 − 𝑧1 𝑒2 = 𝑥0 − 𝑥1 𝑒1 − 𝑥2 𝑒2 + 𝑥3 𝑒3 , 𝑝(11) = 𝑧2 − 𝑧3 𝑒2 = 𝑦0 + 𝑦1 𝑒1 − 𝑦2 𝑒2 − 𝑦3 𝑒3 , 𝑞 = 𝑥0 + 𝑥1 𝑒1 + 𝑥2 𝑒2 + 𝑥3 𝑒3 , (4) where 𝑥𝑚 ∈ R (𝑚 = 0, 1, 2, 3) and 𝑒𝑟 (𝑟 = 1, 2, 3) are split quaternionic units satisfying noncommutative multiplication rules (for split quaternions, see [1]): 𝑒22 = 𝑒32 = 1, 𝑒1 𝑒2 = −𝑒2 𝑒1 = 𝑒3 , 𝑒2 𝑒3 = −𝑒3 𝑒2 = −𝑒1 , (5) 𝑒3 𝑒1 = −𝑒1 𝑒3 = 𝑒2 . Similarly, a dual split quaternion 𝑧 can be written as D (S) = {𝑧 | 𝑧 = 𝑝0 + 𝜀𝑝1 , 𝑝𝑟 ∈ S, 𝑟 = 0, 1} , 𝑝(13) = 𝑧2 − 𝑧3 𝑒2 = 𝑦0 − 𝑦1 𝑒1 − 𝑦2 𝑒2 + 𝑦3 𝑒3 , with 𝑧0 = 𝑥0 − 𝑥1 𝑒1 , 𝑧1 = 𝑥2 − 𝑥3 𝑒1 , 𝑧2 = 𝑦0 − 𝑦1 𝑒1 , and 𝑧3 = 𝑦2 − 𝑦3 𝑒1 . For instance, (6) = (𝑥0 − 𝑥1 𝑒1 + 𝑥2 𝑒2 − 𝑥3 𝑒3 ) 𝑒2 = 𝑝(02) 𝑒2 , 𝑒1 𝑝1 = 𝑒1 (𝑦0 + 𝑦1 𝑒1 + 𝑦2 𝑒2 + 𝑦3 𝑒3 ) Because of the properties of the eight-unit equality, the addition and subtraction of dual split quaternions are governed by the rules of ordinary algebra. Here the symbol 𝑝(𝑘𝑟) is used by just enumerating 𝑟 and 𝑘, not 𝑟 times 𝑘. For example, 𝑝(22) ≠ 𝑝4 and 𝑝22 = 𝑝4 . For any two elements 𝑧 = 𝑝0 + 𝜀𝑝1 and 𝑤 = 𝑞0 + 𝜀𝑞1 of D(S), where 𝑞0 = ∑3𝑟=0 𝑠𝑟 𝑒𝑟 and 𝑞1 = ∑3𝑟=0 𝑡𝑟 𝑒𝑟 are split quaternion components and 𝑠𝑟 , 𝑡𝑟 ∈ R (𝑟 = 0, 1, 2, 3), their noncommutative product is given by 𝑧𝑤 = (𝑝0 + 𝜀𝑝1 ) (𝑞0 + 𝜀𝑞1 ) = 𝑝0 𝑞0 + 𝜀 (𝑝0 𝑞1 + 𝑝1 𝑞0 ) . = (𝑧0 + 𝑧1 𝑒2 ) + 𝜀 (𝑧2 + 𝑧3 𝑒2 ) (7) = (𝑧0 𝑧0 − 𝑧1 𝑧1 ) + 2𝜀 (𝑧0 𝑧2 − 𝑧1 𝑧3 ) where 𝑝0 = 𝑧0 + 𝑧1 𝑒2 and 𝑝1 = 𝑧2 + 𝑧3 𝑒2 are split quaternion components, 𝑧0 = 𝑥0 + 𝑥1 𝑒1 , 𝑧1 = 𝑥2 + 𝑥3 𝑒1 , 𝑧2 = 𝑦0 + 𝑦1 𝑒1 , and 𝑧3 = 𝑦2 + 𝑦3 𝑒1 are usual complex numbers, and 𝑥𝑚 , 𝑦𝑚 ∈ R (𝑚 = 0, 1, 2, 3). The multiplication of split quaternionic units with a dual symbol is commutative 𝜀𝑒𝑟 = 𝑒𝑟 𝜀 (𝑟 = 1, 2, 3). However, by properties of split quaternionic unit, (𝑘 = 0, 1, 2, 3, 𝑟 = 0, 1) , 𝑧𝑘 𝑒𝑟 = 𝑒𝑟 𝑧𝑘 (𝑘 = 0, 1, 2, 3, 𝑟 = 2, 3) , 𝑒𝑟 𝑝𝑘 = 𝑝(𝑘𝑟) 𝑒𝑟 𝑧∗ = 𝑝0∗ + 𝜀𝑝1∗ , 𝑧𝑧∗ = 𝑧∗ 𝑧 = 𝑝0 𝑝0∗ + 𝜀 (𝑝0 𝑝1∗ + 𝑝1 𝑝0∗ ) = 𝑝0 + 𝜀𝑝1 , 𝑧𝑘 𝑒𝑟 = 𝑒𝑟 𝑧𝑘 (𝑟 = 1, 2, 3, 𝑘 = 0, 1) , (11) The conjugation 𝑧∗ of 𝑧 and the corresponding modulus 𝑧𝑧∗ in D(S) are defined by 𝑧 = {(𝑥0 + 𝑥1 𝑒1 ) + (𝑥2 + 𝑥3 𝑒1 ) 𝑒2 } + 𝜀 {(𝑦0 + 𝑦1 𝑒1 ) + (𝑦2 + 𝑦3 𝑒1 ) 𝑒2 } (10) = (𝑦0 + 𝑦1 𝑒1 − 𝑦2 𝑒2 − 𝑦3 𝑒3 ) 𝑒1 = 𝑝(11) 𝑒1 . which has elements of the following form: 𝑒𝑟 𝑝𝑘 ≠ 𝑝𝑘 𝑒𝑟 , (9) 𝑒2 𝑝0 = 𝑒2 (𝑥0 + 𝑥1 𝑒1 + 𝑥2 𝑒2 + 𝑥3 𝑒3 ) and a split quaternion 𝑞 ∈ S is an expression of the form 𝑒12 = −1, 𝑝(01) = 𝑧0 − 𝑧1 𝑒2 = 𝑥0 + 𝑥1 𝑒1 − 𝑥2 𝑒2 − 𝑥3 𝑒3 , 𝑝(12) = 𝑧2 + 𝑧3 𝑒2 = 𝑦0 − 𝑦1 𝑒1 + 𝑦2 𝑒2 − 𝑦3 𝑒3 , 2. Preliminaries 𝜀 ≠ 0, where (12) 1 2 = ∑ {(𝑥𝑟2 − 𝑥𝑟+2 ) + 𝜀 (𝑥𝑟 𝑦𝑟 − 𝑥𝑟+2 𝑦𝑟+2 )} , 𝑟=0 where 𝑝0∗ = 𝑧0 − 𝑧1 𝑒2 and 𝑝1∗ = 𝑧2 − 𝑧3 𝑒2 . Lemma 1. For all 𝑧 ∈ D(S) and 𝑛 ∈ N := {1, 2, 3, . . .} (...truncated)