SPLIT HYPERHOLOMORPHIC FUNCTION IN CLIFFORD ANALYSIS

Abstract

We define a hyperholomorphic function with values in split quaternions, provide split hyperholomorphic mappings on ${\Omega}{\subset}\mathbb{C}^2$ and research the properties of split hyperholomorphic functions.

1. INTRODUCTION

A set of quaternions can be represented as

where and e1e2e3 = −1, which is non-commutative division algebra. A set of split quaternions can be expressed as

where and e1e2e3 = 1, which is also non-commutative. On the other hand, unlike quaternion algebra, a set of split quaternions contains zero divisors, nilpotent elements and non-trivial idempotents. Because split quaternions are used to express Lorentzian rotations, studies of the geometric and physical appli-cations of split quaternions require solving split quaternionic equations (see [6], [9]). Deavours [3] generated regular functions in a quaternion analysis and provided the Cauchy-Fueter integral formulas for regular quaternion functions. Carmody [1, 2] investigated the properties of hyperbolic quaternions, octonions, and sedenions, and Sangwine and Bihan [10] provided a new polar representation of quaternions that is represented by a pair of complex numbers in the Cayley-Dickson form.

We shall denote by and , respectively, the field of complex numbers and the field of real numbers. We [4, 5] showed that any complex-valued harmonic function f1 in a pseudoconvex domain D of 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 method. We define a split hyperholomorphic function with values in split quaternions and examine the properties of split hyperholomorphic functions based on [7].

 

2. PRELIMINARY

The split quaternionic field S is a four-dimensional non-commutative -field generated by four base elements e0, e1, e2, and e3 with the following non-commutative multiplication rules :

The element e0 is the identity of S, and e1 identifies the imaginary unit in the -field of complex numbers. A split quaternion z is given by

where z1 = x0+e1x1, z2 = x2+e1x3, and are complex numbers in and xk (k = 0, 1, 2, 3) are real numbers.

The multiplications of two pure split quaternions and is defined as follows:

For pure split quaternions , and , the cross product satisfies two rules as follows:

The split quaternionic conjugate z*, the multiplicative modulus M(z) and the inverse z−1 of z in S are defined as

We let

The split quaternion number z of S is

z = ξ0 + Jξ1,

where ξ0 = x0 and Then the split quaternionic conjugate number of z is z* =ξ0 − Jξ1, and the multiplicative modulus of z is

Let Ω be an open set in and consider a function f defined on Ω­ with values in S:

where u = u0 and with

We give differential operators as

where and

where Then the Coulomb operator (see [8]) is

Definition 2.1. Let Ω­ be an open set in . A function f(z) = f1(z) + f2(z)e2 is said to be an L(R)-split hyperholomorphic function on Ω if the following two conditions are satisfied:

(1) f1(z) and f2(z) are continuously differential functions on Ω ­, and (2) D*f(z) = 0 (f(z)D* = 0) on Ω­.

In this paper, we consider a L-split hyperholomorphic function on Ω in .

 

3. SPLIT HYPERHOLOMORPHIC FUNCTION

Let ξ0 = r cosh θ and ξ1 = r sinh θ with r2 = |zz*|. Then any z = ξ0 +Jξ1 can be expressed as z = r(cosh θ + J sinh θ), where θ is the angle between the vector and the real axis.

Theorem 3.1. Let ­Ω be a domain of holomorphy in . If u(r, θ) is a split quaternion function satisfying M(D)f = 0 on Ω­, then there exists a split hyper-conjugate quaternion function v(r, θ) satisfying M(D)f = 0 such that u(r, θ) + Jv(r, θ) is a split hyperholomorphic function on Ω.

Proof. We put

We operate the operator ∂ from the left-hand side of ϕ (r, θ) on Ω.

Since and , we get ∂ϕ(r, θ) is zero. Since Ω is a domain of holomorphy, the ∂-closed form ϕ(r, θ) is a ∂-exact form on Ω­. Hence, there exists a split hyper-conjugate quaternion function v(r, θ) satisfying M(D)f = 0 on Ω such that u(r, θ) + Jv(r, θ) is a split hyperholomorphic function on Ω.

Example 3.2. If the split quaternion function

in a domain of holomorphy is known, then a split hyper-conjugate quaternion function v(r, θ) of u(r, θ) on Ω can be found. That is,

and f(r, θ) = u(r, θ) + Jv(r, θ) is a split hyperholomorphic function satisfying M(D)f = 0 on Ω.

Theorem 3.3. Let Ω be an open set in and f be a split quaternion function satisfying M(D)f = 0 on Ω. Then the multiplicative modulus of D f is

Proof. For f = u + Jv and

where Since and we have

Theorem 3.4. Let f : → be a polar coordinates mapping defined by f(r, θ) = (r cosh θ, r sinh θ). Then the determinant of this mapping is

where

Proof. The chain rule gives

Then

Theorem 3.5. Let f : → be a polar coordinates mapping defined by f(r, θ) = (er cosh θ, er sinh θ). Then the determinant of this mapping is

Proof. We can prove as above Theorem 3.4.

Theorem 3.6. Let Ω be an open set in and f be a split hyperholomorphic function on Ω­. Then there exists a differentiable function φ on Ω such that the vector field

Proof. We let any point on Ω. Consider

where μ(ξ1) is a split quaternion-valued function. By the fundamental theorem of calculus, we can find

Since f is a split hyperholomorphic function on Ω and differentiating with respect to ξ1, we obtain

where and Putting and then we have