We define a hyperholomorphic function with values in split quaternions, provide split hyperholomorphic mappings on
and research the properties of split hyperholomorphic functions.
1. INTRODUCTION
A set of quaternions can be represented as
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]
.
2. PRELIMINARY
The split quaternionic field
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
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
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:
-
(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
.
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
+
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) = (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
(
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
Kajiwara J.
,
Li X.D.
,
Shon K.H.
2004
International Colloquium on Finite or Infinite Dimensional Complex Analysis and its Applications
Kluwer Academic Publishers
Vietnam
Regeneration in complex, quaternion and Clifford analysis
Kajiwara J.
,
Li X.D.
,
Shon K.H.
2006
International Colloquium on Finite or Infinite Dimensional Complex Analysis and its Applications
Hue University
Vietnam
Function spaces in complex and Clifford analysis
Lang S.
1987
Calculus of several variables
Springer-Verlag
New York
Obolashvili E.
1996
Some partial differential equations in Clifford analysis
Banach Center Publ.
37
(1)
173 -
179
Sangwine S.J.
,
Bihan N.L.
2010
Quaternion polar representation with a complex modulus and complex argument inspired by the Cayley-Dickson form
Adv. in Appl. Cliff. Algs.
20
(1)
111 -
120
DOI : 10.1007/s00006-008-0128-1