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

(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

(2)

where

, and e₁e₂e₃ = 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 eight-dimensional 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 Pennestrì 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 + f₂j 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 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 Cauchy-Riemann systems and expressions of power series in dual split quaternions from the definition of dual split regular on an open set Ω ⊂ ℂ² × ℂ².

2. Preliminaries

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

(3)

and a split quaternion q ∈ 𝒮 is an expression of the form

(4)

where x_m ∈ ℝ (m = 0,1, 2,3) and e_r (r = 1,2, 3) are split quaternionic units satisfying noncommutative multiplication rules (for split quaternions, see [1]):

(5)

Similarly, a dual split quaternion z can be written as

(6)

which has elements of the following form:

(7)

where p₀ = z₀ + z₁e₂ and p₁ = z₂ + z₃e₂ are split quaternion components, z₀ = x₀ + x₁e₁, z₁ = x₂ + x₃e₁, z₂ = y₀ + y₁e₁, and z₃ = y₂ + y₃e₁ are usual complex numbers, and x_m, y_m ∈ ℝ (m = 0,1, 2,3). The multiplication of split quaternionic units with a dual symbol is commutative εe_r = e_rε (r = 1,2, 3). However, by properties of split quaternionic unit,

(8)

where

(9)

with

, and

. For instance,

(10)

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

and

are split quaternion components and s_r, t_r ∈ ℝ (r = 0,1, 2,3), their noncommutative product is given by

(11)

The conjugation z^* of z and the corresponding modulus zz^* in 𝒟(𝒮) are defined by

(12)

where

and

Lemma 1. For all z ∈ 𝒟(𝒮) and n ∈ ℕ : = {1,2, 3, …}, we have

(13)

Proof. If n = 1, then (13) is trivial. Now suppose that this holds for some n ∈ ℕ. Then, as desired,

(14)

By the principle of mathematical induction, (13) holds for all n ∈ ℕ.

Let Ω be an open subset of ℂ² × ℂ². Then the function f : Ω → 𝒟(𝒮) can be expressed as

(15)

where the component functions f_r : Ω → 𝒮 (r = 0,1) are split quaternionic-valued functions. The component functions f_r (r = 0,1) are

(16)

where g_k = u_2k + u_2k+1e₁ (k = 0,1) and g_k = v_2k−4 + v_2k−3e₁ (k = 2,3) are complex-valued functions, and u_r and v_r (r = 0,1, 2,3) are real-valued functions.

Now, we let differential operators D₁ and D₂ be defined on 𝒟(𝒮) as

(17)

Then the conjugate operators

and

are

(18)

where

(19)

(20)

act on 𝒟(𝒮). These operators are called corresponding Cauchy-Riemann operators in 𝒟(𝒮), where ∂/∂z_r and

are usual differential operators used in the complex analysis.

Remark 2. From the definition of differential operators on 𝒟(𝒮),

(21)

where r = 1,2.

Definition 3. Let Ω be an open set in ℂ² × ℂ². A function f = f₀ + εf₁ is called an L_r (resp., R_r)-regular function (r = 1,2) on Ω if the following two conditions are satisfied:

(i)
f_k (k = 0,1) are continuously differential functions on Ω, and
(ii)
(resp., ) on Ω (r = 1,2).

In particular, the equation

of Definition 3 is equivalent to

(22)

In addition,

(23)

Concretely, the following system is obtained:

(24)

The above systems (23) and (24) are corresponding Cauchy-Riemann systems in 𝒟(𝒮). Similarly, the equation

of Definition 3 is equivalent to

(25)

Then,

(26)

Concretely, the following system is obtained:

(27)

The above systems (26) and (27) are corresponding Cauchy-Riemann systems in 𝒟(𝒮).

On the other hand, the equation

of Definition 3 is equivalent to

(28)

Then,

(29)

where

(30)

Concretely, the following system is obtained:

(31)

Similarly, the equation

of Definition 3 is equivalent to

(32)

Then,

(33)

where

(34)

Concretely, the system is obtained as follows:

(35)

From the systems (24), (27), (31), and (35), the equations

and

are different. Therefore, the equations

and

should be distinguished as L_r-regular functions (r = 1,2) and R_r-regular functions (r = 1,2) on Ω, respectively. Now the properties of the L_r-regular function (r = 1,2) with values in 𝒟(𝒮) are considered.

3. Properties of L_r-Regular Functions (r = 1,2) with Values in 𝒟(𝒮)

We consider properties of a L_r-regular functions (r = 1,2) with values in 𝒟(𝒮).

Theorem 4. Let Ω be an open set in ℂ² × ℂ² and let f = f₀ + εf₁ = (g₀ + g₁e₂) + ε(g₂ + g₃e₂) be an L₁-regular function defined on Ω. Then

(36)

Proof. By the system (23), we have

(37)

Therefore, we obtain

(38)

Theorem 5. Let Ω be an open set in ℂ² × ℂ² and f = f₀ + εf₁ = (g₀ + g₁e₂) + ε(g₂ + g₃e₂) be an L₂-regular function defined on Ω. Then

(39)

Proof. By the system (26), we have

(40)

Therefore, we obtain the following equation:

(41)

Proposition 6. From properties of differential operators, the following equations are obtained:

(42)

Proof. By properties of the power of dual split quaternions and derivatives on 𝒟(𝒮), the following derivatives are obtained:

(43)

The other equations are calculated using a similar method, and the above equations are obtained.

Theorem 7. Let Ω be an open set in ℂ² × ℂ² and let f(z) be a function on Ω with values in 𝒟(𝒮). Then the power zⁿ of z in 𝒟(𝒮) is not an L₁-regular function but an L₂-regular function on Ω, where n ∈ ℕ.

Proof. From the definition of the L_r-regular function (r = 1,2) on Ω and Proposition 6, we may consider whether the power zⁿ of z in 𝒟(𝒮) satisfies the equation . Since ,

(44)

Hence, the power zⁿ of z is not L₁-regular on Ω. On the other hand, from the equations in Proposition 6, we have

, and

. Then,

(45)

Therefore, by the definition of the L_r-regular function (r = 1,2) on Ω, a power zⁿ of z is L₂-regular on Ω.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and Future Planning (2013R1A1A2008978).

References

1 Kantor I. L. and Solodovnikov A. S., Hypercomplex Numbers: An Elementary Introduction to Algebras, 1989, Springer, New York, NY, USA, MR996029.
10.1007/978-1-4612-3650-4
Google Scholar
2 Cockle J., On systems of algebra involving more than one imaginary, Philosophical Magazine III. (1849) 35, no. 238, 434–435, https://doi.org/10.1080/14786444908646384.
10.1080/14786444908646384
Google Scholar
3 Özdemir M. and Ergin A. A., Rotations with unit timelike quaternions in Minkowski 3-space, Journal of Geometry and Physics. (2006) 56, no. 2, 322–336, https://doi.org/10.1016/j.geomphys.2005.02.004, MR2173900.
10.1016/j.geomphys.2005.02.004
Web of Science® Google Scholar
4 Kula L. and Yayli Y., Split quaternions and rotations in semi Euclidean space , Journal of the Korean Mathematical Society. (2007) 44, no. 6, 1313–1327, https://doi.org/10.4134/JKMS.2007.44.6.1313, MR2358956, ZBL1140.15016.
10.4134/JKMS.2007.44.6.1313
Web of Science® Google Scholar
5 Brody D. C. and Graefe E.-M., On complexified mechanics and coquaternions, Journal of Physics A: Mathematical and Theoretical. (2011) 44, no. 7, article 072001, https://doi.org/10.1088/1751-8113/44/7/072001, MR2769792, ZBL1208.81073.
10.1088/1751-8113/44/7/072001
Web of Science® Google Scholar
6 Frenkel I. and Libine M., Split quaternionic analysis and separation of the series for SL(2, ℝ) and SL(2, ℂ)/SL(2, ℝ), Advances in Mathematics. (2011) 228, no. 2, 678–763, https://doi.org/10.1016/j.aim.2011.06.001, MR2822209, ZBL1264.30037.
10.1016/j.aim.2011.06.001
Web of Science® Google Scholar
7 Inoguchi J.-i., Timelike surfaces of constant mean curvature in Minkowski 3-space, Tokyo Journal of Mathematics. (1998) 21, no. 1, 141–152, https://doi.org/10.3836/tjm/1270041992, MR1630159.
10.3836/tjm/1270041992
Google Scholar
8 Kenwright B., A beginners guide to dual-quaternions: what they are, how they work, and how to use them for 3D character hierarchies, Proceedings of the 20th International Conferences on Computer Graphics, Visualization and Computer Vision, 2012, 1–10.
Google Scholar
9 Pennestrì E. and Stefanelli R., Linear algebra and numerical algorithms using dual numbers, Multibody System Dynamics. (2007) 18, no. 3, 323–344, https://doi.org/10.1007/s11044-007-9088-9, MR2355249, ZBL1128.70002.
10.1007/s11044-007-9088-9
Web of Science® Google Scholar
10 Son L. H., An extension problem for solutions of partial differential equations in ℝⁿ, Complex Variables: Theory and Application. (1990) 15, no. 2, 87–92, MR1058516.
Google Scholar
11 Son L. H., Extension problem for functions with values in a Clifford algebra, Archiv der Mathematik. (1990) 55, no. 2, 146–150, https://doi.org/10.1007/BF01189134, MR1064381.
10.1007/BF01189134
Web of Science® Google Scholar
12 Kajiwara J., Li X. D., and Shon K. H., Regeneration in complex, quaternion and Clifford analysis, International Colloquium on Finite or Infinite DimensionalComplex Analysis and Its Applications, 2004, 2, Kluwer Academic Publishers, Hanoi, Vietnam, 287–298, Advances in Complex Analysis and Its Applications no. 9.
Google Scholar
13 Kajiwara J., Li X. D., and Shon K. H., Function spaces in complex and Clifford analysis, International Colloquium on Finite Or Infinite Dimensional Complex Analysis and Its Applications, 2006, 14, Hue University, Hue, Vietnam, 127–155, Inhomogeneous Cauchy Riemann System of Quaternion and Clifford Analysis in Ellipsoid.
Google Scholar
14 Kim J. E., Lim S. J., and Shon K. H., Regular functions with values in ternary number system on the complex Clifford analysis, Abstract and Applied Analysis. (2013) 2013, 7, 136120, MR3147805, https://doi.org/10.1155/2013/136120.
10.1155/2013/136120
Google Scholar
15 Kim J. E., Lim S. J., and Shon K. H., Regularity of functions on the reduced quaternion field in Clifford analysis, Abstract and Applied Analysis. (2014) 2014, 8, 654798, https://doi.org/10.1155/2014/654798, MR3186975.
10.1155/2014/654798
Google Scholar
16 Lim S. J. and Shon K. H., Hyperholomorphic fucntions and hyperconjugate harmonic functions of octonion variables, Journal of Inequalities and Applications. (2013) 2013, article 77, https://doi.org/10.1186/1029-242X-2013-77.
10.1186/1029-242X-2013-77
Google Scholar
17 Lim S. J. and Shon K. H., Dual quaternion functions and its applications, Journal of Applied Mathematics. (2013) 2013, 6, 583813, MR3100823, https://doi.org/10.1155/2013/583813.
10.1155/2013/583813
Google Scholar
18 Kim J. E., Lim S. J., and Shon K. H., Taylor series of functions with values in dual quaternion, Journal of the Korean Society of Mathematical Education B—The Pure and Applied Mathematics. (2013) 20, no. 4, 251–258, MR3154741, ZBL06271632.
Google Scholar

Citing Literature

All articles

The Regularity of Functions on Dual Split Quaternions in Clifford Analysis

Abstract

1. Introduction

2. Preliminaries

3. Properties of L_r-Regular Functions (r = 1,2) with Values in 𝒟(𝒮)

Conflict of Interests

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

The Regularity of Functions on Dual Split Quaternions in Clifford Analysis

Abstract

1. Introduction

2. Preliminaries

3. Properties of Lr-Regular Functions (r = 1,2) with Values in 𝒟(𝒮)

Conflict of Interests

Acknowledgments

References

Citing Literature

References

Related

Information

3. Properties of L_r-Regular Functions (r = 1,2) with Values in 𝒟(𝒮)