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

- Received : October 14, 2014
- Accepted : November 22, 2014
- Published : February 28, 2015

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We define a hyperholomorphic function with values in split quaternions, provide split hyperholomorphic mappings on
and research the properties of split hyperholomorphic functions.
where
and
e
_{1}
e
_{2}
e
_{3}
= −1, which is non-commutative division algebra. A set of split quaternions can be expressed as
where
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
and
, 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
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]
.
S
is a four-dimensional non-commutative
-field generated by four base elements
e
_{0}
,
e
_{1}
,
e
_{2}
, and
e
_{3}
with the following non-commutative multiplication rules :
The element
e
_{0}
is the identity of
S
, and
e
_{1}
identifies the imaginary unit
in the
-field of complex numbers. A split quaternion
z
is given by
where
z
_{1}
=
x
_{0}
+
e
_{1}
x
_{1}
,
z
_{2}
=
x
_{2}
+
e
_{1}
x
_{3}
,
and
are complex numbers in
and
x_{k}
(
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
where
ξ
_{0}
=
x
_{0}
and
Then the split quaternionic conjugate number of
z
is
z
* =
ξ
_{0}
−
Jξ
_{1}
, 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
=
u
_{0}
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
) =
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:
In this paper, we consider a L-split hyperholomorphic function on Ω in
.
ξ
_{0}
=
r
cosh
θ
and
ξ
_{1}
=
r
sinh
θ
with
r
^{2}
= |
zz
*|. Then any
z
=
ξ
_{0}
+
Jξ
_{1}
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. T
hen 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 ) = (e^{r} cosh θ , e^{r} 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
(
ξ
_{1}
) 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
ξ
_{1}
, we obtain
where
and
Putting
and then we have

1. INTRODUCTION

A set of quaternions can be represented as
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2. PRELIMINARY

The split quaternionic field
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- z=ξ0+Jξ1,

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- (1)f1(z) andf2(z) are continuously differential functions on Ω , and
- (2)D*f(z) = 0 (f(z)D* = 0) on Ω.

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3. SPLIT HYPERHOLOMORPHIC FUNCTION

Let
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Carmody K.
1988
Circular and hyperbolic quaternions, octonions and sedenions
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** DOI : 10.1016/0096-3003(88)90133-6**

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Lang S.
1987
Calculus of several variables
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Obolashvili E.
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Özdemir M.
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Citing 'SPLIT HYPERHOLOMORPHIC FUNCTION IN CLIFFORD ANALYSIS
'

@article{ SHGHCX_2015_v22n1_57}
,title={SPLIT HYPERHOLOMORPHIC FUNCTION IN CLIFFORD ANALYSIS}
,volume={1}
, url={http://dx.doi.org/10.7468/jksmeb.2015.22.1.57}, DOI={10.7468/jksmeb.2015.22.1.57}
, number= {1}
, journal={Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={LIM, SU JIN
and
SHON, KWANG GH}
, year={2015}
, month={Feb}