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SPLIT HYPERHOLOMORPHIC FUNCTION IN CLIFFORD ANALYSIS
SPLIT HYPERHOLOMORPHIC FUNCTION IN CLIFFORD ANALYSIS
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Feb, 22(1): 57-63
Copyright © 2015, Korean Society of Mathematical Education
  • Received : October 14, 2014
  • Accepted : November 22, 2014
  • Published : February 28, 2015
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About the Authors
SU JIN LIM
DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY, BUSAN 609-703, KOREAEmail address:sjlim@pusan.ac.kr
KWANG GH SHON
DEPARTMENT OF MATHEMATICS, PUSAN NATIONAL UNIVERSITY, BUSAN 609-703, KOREAEmail address:khshon@pusan.ac.kr

Abstract
We define a hyperholomorphic function with values in split quaternions, provide split hyperholomorphic mappings on ­ and research the properties of split hyperholomorphic functions.
Keywords
1. INTRODUCTION
A set of quaternions can be represented as
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where
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and e 1 e 2 e 3 = −1, which is non-commutative division algebra. A set of split quaternions can be expressed as
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where
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and e 1 e 2 e 3 = 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
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and
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, respectively, the field of complex numbers and the field of real numbers. We [4 , 5] showed that any complex-valued harmonic function f 1 in a pseudoconvex domain D of
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has a conjugate function f 2 in D such that the quaternion-valued function f 1 + f 2 j 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
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-field generated by four base elements e 0 , e 1 , e 2 , and e 3 with the following non-commutative multiplication rules :
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The element e 0 is the identity of S , and e 1 identifies the imaginary unit
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in the
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-field of complex numbers. A split quaternion z is given by
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where z 1 = x 0 + e 1 x 1 , z 2 = x 2 + e 1 x 3 ,
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and
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are complex numbers in
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and xk ( k = 0, 1, 2, 3) are real numbers.
The multiplications of two pure split quaternions
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and
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is defined as follows:
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For pure split quaternions
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,
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and
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, the cross product satisfies two rules as follows:
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The split quaternionic conjugate z *, the multiplicative modulus M ( z ) and the inverse z −1 of z in S are defined as
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We let
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The split quaternion number z of S is
  • z=ξ0+Jξ1,
where ξ 0 = x 0 and
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Then the split quaternionic conjugate number of z is z * = ξ 0 1 , and the multiplicative modulus of z is
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Let Ω be an open set in
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and consider a function f defined on Ω­ with values in S :
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where u = u 0 and
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with
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We give differential operators as
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where
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and
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where
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Then the Coulomb operator (see [8] ) is
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Definition 2.1. Let Ω­ be an open set in
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. A function f ( z ) = f 1 ( z ) + f 2 ( z ) e 2 is said to be an L ( R )- split hyperholomorphic function on Ω if the following two conditions are satisfied:
  • (1)f1(z) andf2(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
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.
3. SPLIT HYPERHOLOMORPHIC FUNCTION
Let ξ 0 = r cosh θ and ξ 1 = r sinh θ with r 2 = | zz *|. Then any z = ξ 0 + 1 can be expressed as z = r (cosh θ + J sinh θ ), where θ is the angle between the vector
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and the real axis.
Theorem 3.1. Let ­Ω be a domain of holomorphy in
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. 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
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We operate the operator ∂ from the left-hand side of ϕ ( r , θ ) on Ω.
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Since
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and
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, 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.
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Example 3.2. If the split quaternion function
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in a domain of holomorphy
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is known, then a split hyper-conjugate quaternion function v ( r , θ ) of u ( r , θ ) on Ω can be found. That is,
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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
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and f be a split quaternion function satisfying M ( D ) f = 0 on Ω. T hen the multiplicative modulus of D f is
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Proof. For f = u + Jv and
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where
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Since
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and
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we have
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Theorem 3.4. Let f :
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be a polar coordinates mapping defined by f(r, θ) = (r cosh θ, r sinh θ). Then the determinant of this mapping is
where
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Proof. The chain rule gives
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Then
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Theorem 3.5. Let f :
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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.
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Theorem 3.6. Let Ω be an open set in
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and f be a split hyperholomorphic function on Ω­. Then there exists a differentiable function φ on Ω such that the vector field
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Proof. We let any point
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on Ω. Consider
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where μ ( ξ 1 ) is a split quaternion-valued function. By the fundamental theorem of calculus, we can find
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Since f is a split hyperholomorphic function on Ω and differentiating with respect to ξ 1 , we obtain
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where
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and
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Putting
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and then we have
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