Kim Advances in Difference Equations (2017) 2017:80
DOI 10.1186/s13662-017-1131-4
RESEARCH
Open Access
A corresponding Cullen-regularity for
split-quaternionic-valued functions
Ji Eun Kim*
*
Correspondence:
[email protected]
Department of Mathematics,
Dongguk University, Gyeongju-si,
38066, Republic of Korea
Abstract
We give a representation of the class of Cullen-regular functions in split-quaternions.
We consider each Cullen’s form of split-quaternions, which provides corresponding
Cauchy-Riemann equations for split-quaternionic variables. Using Cullen’s form, we
research hyperholomorphy and the properties of functions of split-quaternionic
variables which are expressed by hyperbolic coordinates.
MSC: Primary 32A99; secondary 32W50; 30G35; 11E88
Keywords: split-quaternion; Cullen’s form; regular function; Cauchy-Riemann
system; Clifford analysis
1 Introduction
The skew field of real quaternions, denoted by H, has the form
H = q | q = x + ix + jx + kx , xr ∈ R (r = , , , ) ,
where R is the set of real numbers and i, j, and k are imaginary units with
i = j = k  = –,
ij = –ji = k,
jk = –kj = i,
ki = –ik = j.
Theories and applications of functions of a quaternionic variable have been led by holomorphic functions of one complex variable. Quaternions were introduced by Hamilton
[] in  and generalized by Clifford [] in . After then, Fueter [, ] defined differential operators, called the Cauchy-Fueter operators, and regular functions in the space
of solutions of the equation with these operators. Kim et al. [, ] obtained the regularity
of functions on the form of reduced quaternions in Clifford analysis.
Split-quaternions are elements of four-dimensional algebra introduced by Cockle []
in . Similar to quaternions, they form a four-dimensional real vector space which is a
general associative, but non-commutative, form of multiplication. Unlike the quaternions,
the split-quaternions contain zero divisors, nilpotent elements, and nontrivial idempois an idempotent zero-divisor, and i – j is nilpotent. Indeed, Helmut and
tents such as (+j)

Kist [] constructed an algebra over the real numbers, which is isomorphic to the algebra
of  ×  real matrices. Moreover, the modulus which has an isotropic quadratic form for
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Kim Advances in Difference Equations (2017) 2017:80
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split-quaternions provides hyperbolic motions of the Poincaré disk model of hyperbolic
geometry. From this result, the structure of split-quaternion analysis has been developed
and applied in four-dimensional physics by Frenkel and Libine [, ]. Kim and Shon [,
] researched corresponding Cauchy-Riemann systems and the properties of hyperholomorphic functions with values in modified split-quaternions. Kim and Shon [] gave relations between the properties of split-quaternionic regular functions and their algebraic
inverse mapping.
In , Cullen [] proposed the notion of intrinsic functions
and found quaternions
convenient to write in the form ζ = x + pμ, where x and p = x + x + x are real and μ
is a unit vector quaternion. For fixed μ, the elements of the form x + pμ constitute a subspace of D which is isomorphic to the complex field. Since D is a generalization of the algebra of complex numbers, D has properties analogous to the class of analytic functions of
a complex variable. From these properties, Cullen-regular functions are tried to be related
to a class of functions of the reduced quaternionic variable, studied by Leutwiler in [].
Gentili and Struppa [] and Alayón-Solarz [, ] gave definitions of regularity for functions of a quaternionic variable and developed representations of the Cullen-regularity of
quaternion analysis. By using the analytic properties and calculating processes of Cullenregularity, Marin has researched thermoelastic materials in various points of view. For
example, Marin and coauthors studied the asymptotic partition of total energy for the solutions of the mixed initial boundary value problem within the thermoelasticity of initially
stressed bodies. They obtained a spatial decay estimate which is considered a right cylinder composed of physically micropolar thermoelastic material for which one plane end is
subjected to an excitation harmonic in time. Also, they extended the concept of domain
of influence in order to cover the elasticity of microstretch materials and studied it for the
displacement field, the microrotation field, and the microstretch function (see [–]).
Based on these studies, we consider a general type of Cullen-regularity for functions
of split-quaternionic variables. First, we give the notions and some properties of splitquaternion-valued functions by using Cullen’s form. Also, we investigate the structure of
a Cullen-regular function and corresponding split-Cauchy-Riemann systems for Cullenvariables and research properties of hyperholomorphic functions, represented by Cullenregular functions.
2 Preliminaries
Let S denote the skew field of real split-quaternions which has elements of the form
p = x + ix + jx + kx ,
where xr (r = , , , ) are real, i, j and k are imaginary units such that
i = –,
ij = –ji = k,
j = k  = ,
kj = –jk = i,
ki = –ik = j,
which is isomorphic to R . By the properties of the imaginary units of split-quaternions,
we have the following rules for addition and multiplication:
p + q = (x + y ) + i(x + y ) + j(x + y ) + k(x + y )
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and
pq = (x y – x y + x y + x y ) + i(x y + x y + x y – x y )
+ j(x y + x y + x y – x y ) + k(x y – x y + x y + x y ),
respectively. We give the conjugation of a split-quaternion as follows:
p∗ = x – ix – jx – kx .
Then we have a modulus, denoted by N (p), and an inverse element, denoted by p– of
p ∈ S,
N (p) := pp∗ = x + x – x – x
and
p– =
p∗
N (p)

x + x = x + x ,
respectively. By the non-commutative property of basis vectors i, j and k, the product of
a split-quaternion with its conjugate is given in an isotropic quadratic form. Given two
split-quaternions p and q, the following holds:
N(pq) = N(p)N(q).
For any p =  with N(p) = , p is a null vector, and when the modulus is non-zero, then p
has a multiplicative inverse.
For a split-quaternion p, it can be written as p = S(p) + V (p), where S(p) is the scalar part
and V (p) is the vector part of p. Specially, if S(p) = , then p is called pure split-quaternion.
We focus the attention on the pure split-quaternions to configure Cullen’s form in S. For
a pure split-quaternion p, we consider T = {p = ix + jy + kz | –x + y + z = }. Since points
(x, y, z), which satisfy –x + y + z = , compose Figure  in practice, Figure  shows that T
Figure 1 Figures of –x 2 + y2 + z2 = 1.
Kim Advances in Difference Equations (2017) 2017:80
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Figure 2 x 2 < y2 + z2 .
(a) Onto the xy-plane
(b) Onto the yz-plane
(c) Onto the xz-plane
Figure 3 The projection of Figure 2.
can exist and be non-empty in S. So, we can represent the following processes and results.
If we let
ix + jx + kx
J=
–x + x + x
x < x + x ,
then J  =  and J ∈ T . Indeed, the existence of the element J of T is guaranteed by Figures -.
Let be an open set in S. A function f : → S is given by
f (p) = f + if + jf + kf ,
where fr = fr (x , x , x , x ) (r = , , , ) are real-valued functions. We try to express a function using the element J of T . So, we consider the class of split-quaternionic functions
Kim Advances in Difference Equations (2017) 2017:80
(a) Figure of T
Page 5 of 14
(b) Figure of n(p) on the xy-plane of T
Figure 4 Figure of n(p) on T .
which has an element f written as
f = u + Jv,
(.)
by letting u = f be a real-valued function and

v= (–x f + x f + x f ) + i(–x f + x f )
–x + x + x
+ j(–x f + x f ) + k(x f – x f )
be a split-quaternion-valued function. Let J := ∪ LJ , where LJ = R + JR. For all p ∈ J ,
let fJ : J → S be a function
fJ (p) = f (x + Jy) = u(x, y) + Jv(x, y),
(.)
where u and v are real-valued functions. The function fJ is called a restriction of f in S.
Remark . For any split-quaternion p ∈ S, the identity function maps one split-quaternion onto itself, and there are unique x, y ∈ R with y > , and J ∈ T such that p = x + Jy,
where
x = x ,
y=
–x + x + x ,
ix + jx + kx
J=
–x + x + x
x < x + x .
3 Regularity of split-quaternionic functions
Consider differential operators given by
D∗l :=
∂
∂
∂
∂
+i
–j
–k
∂x
∂x
∂x
∂x
(.)
D∗r :=
∂
∂
∂
∂
+
i–
j–
k,
∂x ∂x
∂x
∂x
(.)
and
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where the units i, j and k act from the left or right in each case. We call (.) the leftdifferential operator and (.) the right-differential operator in S. Since both operators
have similar roles, we will describe D∗l only later.
Definition . Let be an open set in S. A function f : → S is said to be J-regular if for
every J ∈ T , its restriction fJ : J → S is continuously differentiable and satisfies
D∗J f (x + Jy) = ,
where
D∗J
∂
 ∂
–J
.
:=
 ∂x
∂y
Remark . The equation D∗J f (x + Jy) =  is equivalent to the following equations:
∂u
∂x
∂v
∂x
–
–
∂v
∂y
∂u
∂y
= ,
= .
(.)
So, a function f is J-regular if and only if f satisfies equation (.).
Example  For p ∈ S and an ∈ S (n ∈ N, where N is the set of positive integers), a polynomial f (p) = pn an satisfies
D∗J f (p) =
∂
 ∂
–J
(x + Jy)n an = .
 ∂x
∂y
Hence, the polynomial pn an is J-regular.
Polynomials are used in a wide variety of fields of mathematics and every branch of
chemistry and physics. In advanced mathematics, polynomials are used to construct polynomial rings and central concepts in algebra and algebraic geometry. Moreover, if a function is a solution of the equations consisting of differential operators in some systems, the
function can be locally expanded as a power series by Taylor’s theorem.
Proposition . The sum and product of two J-regular functions are J-regular.
Proof Let f = u (x, y) + Jv (x, y) and g = u (x, y) + Jv (x, y) be J-regular. Then D∗J f = D∗J g = .
From the rules of addition and product for p = x + Jy ∈ S, we get
∂  ∂
–J
(u ± u ) + J(v ± v )
 ∂x
∂y

∂u ∂v
∂u ∂v
–
±
–
=

∂x
∂y
∂x
∂y
∂u ∂v
∂u ∂v

+
∓
–
+ J –

∂y
∂x
∂y
∂x
D∗J (f ± g) =
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and
∂  ∂
–J
(u u ) + (v v ) + J(u v + v u )
 ∂x
∂y
∂u ∂v
∂v ∂u
∂u ∂v

–
u + u
–
+
–
v
=

∂x
∂y
∂x
∂y
∂x
∂y
∂v
∂u
– v
+ v
∂x
∂y
∂u ∂v

∂u ∂v
∂v ∂u
+
u – u
–
–
–
v
+ J –

∂y
∂x
∂y
∂x
∂y
∂x
∂v ∂u
–
– v
∂y
∂x
D∗J (fg) =
respectively. Since uλ and vλ (λ = , ) are real-valued functions, by using equation (.),
we obtain D∗J (f ± g) =  and D∗J (fg) = . Therefore, the sum and product of two J-regular
functions are J-regular.
Proposition . If a function is J-regular and non-zero, then its algebraic inverse is Jregular.
Proof Let f = u(x, y) + Jv(x, y) be J-regular. Then its algebraic inverse, denoted by (f )– , is
(f )– =
f∗
u – Jv
=  
N (f ) u – v
u = v .
Since u and v are real-valued functions, we have
D∗J
(f )
–
 ∂
∂
u – Jv
=
–J
 ∂x
∂y
u – v
∂u

∂v   ∂v
∂u
=
–
J
–
v
–
v
–
(u
–
Jv)
u
u
(u – v )
∂x
∂x
∂x
∂x
∂u
∂v
∂u ∂v   +
u – v + J(u – Jv) u
– v
–J
∂y ∂y
∂y
∂y
∂u ∂v
∂v ∂u



(u – Jv) –
+
+ J(u – Jv) – +
.
=
(u – v )
∂x ∂y
∂x ∂y
From equation (.), we obtain D∗J ((f )– ) = . Therefore, the function (f )– is J-regular. Definition . Let be an open set in S. Let a function f : → S be differentiable. Then
DJ f is said to be a J-derivative of f in S by
∂
 ∂
+J
fJ (x + Jy).
DJ f (x + Jy) =
 ∂x
∂y
4 Hyperholomorphy in the coordinate system
The coordinate system used by the variables (t, r, θ , ϕ) can be represented by hyperbolic
coordinates
J = (cosh θ sinh ϕ, sinh θ sinh ϕ, cosh ϕ).
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For a split-quaternion p = t + Jr, that is, p = t + ir cosh θ sinh ϕ + jr sinh θ sinh ϕ + kr cosh ϕ ∈
S, a function f : → S is written as follows:
f (p) = u(t, r, θ , ϕ) + Jv(t, r, θ , ϕ).
We let Jθ and Jϕ be the derivatives of J with respect to θ and ϕ, respectively, such that
Jθ = i sinh θ sinh ϕ + j cosh θ sinh ϕ,
Jϕ = i cosh θ cosh ϕ + j sinh θ cosh ϕ + k sinh ϕ,
Jθ– = i(sinh ϕ)– sinh θ + j(sinh ϕ)– cosh θ
and
Jϕ– = –i cosh θ cosh ϕ – j sinh θ cosh ϕ – k sinh ϕ,
where Jθ– and Jϕ– are algebraic inverse elements of Jθ and Jϕ , respectively, that is, the equations
Jθ Jθ– = Jθ– Jθ = 
and
Jϕ Jϕ– = Jϕ– Jϕ = 
hold.
Lemma . The left-differential operator (.) in this coordinate system has the form
D∗l =

∂
 ∂
∂
–
J +
,
∂t  ∂r r ∂J
where
∂
∂
∂
= (Jθ )–
+ (Jϕ )– .
∂J
∂θ
∂ϕ
Proof From the representation of J, we have Jr = ix + jx + kx . Then
∂
∂
∂
∂
∂
∂
∂
,
= (–Ji)
+ (Jj)
+ (Jk)
= (–J) i
–j
–k
∂r
∂x
∂x
∂x
∂x
∂x
∂x
∂
∂
∂
∂
∂
∂
∂
= (–ri)
+ (rj)
+ (rk)
= (–r) i
–j
–k
∂J
∂x
∂x
∂x
∂x
∂x
∂x
and
∂
∂
= (Jθ ) ,
∂θ
∂J
∂
∂
= (Jϕ ) .
∂ϕ
∂J
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Thus, we obtain
i
∂
 ∂
∂
∂

∂
–J +
.
–j
–k
=
∂x
∂x
∂x 
∂r –r ∂J
This coordinate system can be used to show that a function is hyperholomorphic. Let
p = t + Jr ∈ S (t, r ∈ R) and f (p) = u(t, r, θ , ϕ) + Jv(t, r, θ , ϕ), where u and v are real-valued
functions.
Definition . Let be an open set. Then a function f = u(t, r, θ , ϕ) + Jv(t, r, θ , ϕ) is said
to be hyperholomorphic if u, v ∈ C  () and f satisfies D∗l f = .
Theorem . A function f (p) = u(t, r, θ , ϕ) + Jv(t, r, θ , ϕ) is hyperholomorphic if and only if
u and v satisfy the following equations:
⎧
∂u
 ∂v
⎪
⎨ ∂t –  ∂r = ,
∂v
 ∂u
–  ∂r = ,
∂t
⎪
⎩ ∂u ∂(Jv)
+ ∂J = .
∂J
(.)
More precisely,
⎧ ∂u  ∂v
–  ∂r = ,
⎪
∂t
⎪
⎪
⎨ ∂v –  ∂u = ,
∂t
 ∂r
⎪
+
(sinh ϕ)– ∂u
⎪
∂θ
⎪
⎩
– ∂v
(sinh ϕ) ∂θ +
∂v
∂ϕ
∂u
∂ϕ
=  (or
=  (or
∂u
∂θ
∂v
∂θ
∂v
+ sinh ϕ ∂ϕ
= ),
∂u
+ sinh ϕ ∂ϕ
= ).
Proof From the definition of a hyperholomorphic function in split-quaternions, we have
D∗l f
=
∂
 ∂
 ∂
– J –
(u + Jv)
∂t  ∂r r ∂J
∂v  ∂v  ∂(Jv)
∂u  ∂u  ∂u
– J
–
+J
–
–
∂t  ∂r r ∂J
∂t  ∂r r ∂J
∂v  ∂u
 ∂u ∂(Jv)
∂u  ∂v
–
+J
–
–
+
.
=
∂t  ∂r
∂t  ∂r
r ∂J
∂J
=
Since D∗l f = , we obtain equation (.). Specially, the equation
∂u ∂(Jv)
+
=
∂J
∂J
can be more specifically written as follows.
Since we have
(Jθ )– J = (sinh ϕ)– (i sinh θ + j cosh θ )
× (i cosh θ sinh ϕ + j sinh θ sinh ϕ + k cosh ϕ)
= (sinh ϕ)– (–i cosh θ cosh ϕ – j sinh θ cosh ϕ – k sinh ϕ)
= (sinh ϕ)– (Jϕ )–
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and
(Jϕ )– J = (–i cosh θ cosh ϕ – j sinh θ cosh ϕ – k sinh ϕ)
× (i cosh θ sinh ϕ + j sinh θ sinh ϕ + k cosh ϕ)
= cosh ϕ – sinh ϕ (i sinh θ + j cosh θ )
= sinh ϕ(Jθ )– ,
we get
∂v
∂v
∂u
∂u
∂u ∂(Jv)
+
= (Jθ )–
+ (Jϕ )–
+ (Jθ )– J
+ (Jϕ )– J
∂J
∂J
∂θ
∂ϕ
∂θ
∂ϕ
= (sinh ϕ)– (Jϕ )– J
∂u
∂u
+ (Jϕ )–
∂θ
∂ϕ
∂v
∂v
+ (Jϕ )– J
∂θ
∂ϕ
∂v
–
– ∂u
= (Jϕ ) J (sinh ϕ)
+
∂θ ∂ϕ
∂u
∂v
+ (sinh ϕ)–
+ (Jϕ )–
∂ϕ
∂θ
+ (sinh ϕ)– (Jϕ )–
= .
Thus, we obtain
+
(sinh ϕ)– ∂u
∂θ
∂v
(sinh ϕ)– ∂θ
+
∂v
∂ϕ
∂u
∂ϕ
= ,
= .
Proposition . The sum and product of two hyperholomorphic functions are hyperholomorphic.
Proof Let f = u (t, r, θ , ϕ) + Jv (t, r, θ , ϕ) and g = u (t, r, θ , ϕ) + J(t, r, θ , ϕ) be hyperholomorphic. Then D∗l f = D∗l g = . From the rules of addition and product for p = t + Jr ∈ S with
the coordinate system, we get
 ∂
 ∂ ∂
– J –
(u ± u ) + J(v ± v )
∂t  ∂r r ∂J
 ∂
 ∂ ∂
– J
(u ± u ) + J(v ± v ) –
(u ± u ) + J(v ± v )
=
∂t  ∂r
r ∂J
 ∂u ∂u ∂(Jv ) ∂(Jv )
=–
±
+
±
r ∂J
∂J
∂J
∂J
D∗l (f ± g) =
and
D∗l (fg)
 ∂
 ∂ ∂
– J –
(u u ) + (v v ) + J(u v + v u )
=
∂t  ∂r r ∂J
 ∂ ∂
– J
(u u ) + (v v ) + J(u v + v u )
=
∂t  ∂r
Kim Advances in Difference Equations (2017) 2017:80
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 ∂
(u u ) + (v v ) + J(u v + v u )
r ∂J
∂ ∂ 
(Jθ )–
(u u ) + (v v ) + (Jϕ )–
(u u ) + (v v )
=–
r
∂θ
∂ϕ
–
∂ (u v ) + (v u )
∂θ
– ∂
+ (sinh ϕ)(Jθ )
(u v ) + (v u )
∂ϕ
∂v
∂v
∂u
∂u

+ sinh ϕ
+ u
+ sinh ϕ
= – (Jθ )– u
r
∂θ
∂ϕ
∂θ
∂ϕ
∂v
∂v
∂u
∂u
+ v
+ sinh ϕ
+ v
+ sinh ϕ
∂θ
∂ϕ
∂θ
∂ϕ

∂v
∂u
+ (sinh ϕ)–
– (Jϕ )– u
r
∂ϕ
∂θ
∂u
∂v
– ∂v
– ∂u
+ (sinh ϕ)
+ v
+ (sinh ϕ)
+ u
∂ϕ
∂θ
∂ϕ
∂θ
∂v
∂u
+ (sinh ϕ)–
,
+ v
∂ϕ
∂θ
+ (sinh ϕ)– (Jϕ )–
respectively. Since uλ and vλ (λ = , ) are real-valued functions, by using equation (.),
we obtain D∗l (f ± g) =  and D∗l (fg) = . Therefore, the sum and product of two hyperholomorphic functions are hyperholomorphic.
Proposition . If a function is hyperholomorphic and non-zero, then its algebraic inverse
is hyperholomorphic.
Proof Let f = u(t, r, θ , ϕ) + Jv(t, r, θ , ϕ) be hyperholomorphic. Then its algebraic inverse
(f )– =
u – Jv
f∗
=  
N (f ) u – v
u = v ,
where u and v are real-valued functions, satisfies
D∗l
(f )
–
 ∂
 ∂ u – Jv
∂
=
– J –
∂t  ∂r r ∂J u – v
 ∂ u – Jv
∂
 ∂ u – Jv
– J
=
–
∂t  ∂r u – v r ∂J u – v
u
Jv
 ∂
∂
=–
–
r ∂J u – v
∂J u – v
∂u   ∂u

∂v
∂v
= – (Jθ )–
u – v – u
+ uv
– sinh ϕ u
r
∂θ
∂θ
∂θ
∂ϕ
∂v
∂u

– sinh ϕ v + sinh ϕ uv

∂ϕ
∂ϕ
(u – v )
∂v

∂u
∂v
∂u   u – v – u
+ uv
– (sinh ϕ)– u
– (Jϕ )–
r
∂ϕ
∂ϕ
∂ϕ
∂θ

∂v
∂u
– (sinh ϕ)– v + (sinh ϕ)– uv
∂θ
∂θ
(u – v )
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 ∂u

∂v
–
= – (Jθ )
+ (sinh ϕ)
–u
r
∂θ
∂ϕ
∂u
∂v
∂v
∂u
+ –v
+ (sinh ϕ)
+ (uv)
+ (sinh ϕ)
∂θ
∂ϕ
∂θ
∂ϕ
∂u

∂v
+ (sinh ϕ)–
– (Jϕ )– –u
r
∂ϕ
∂θ
∂u
∂v
+ (sinh ϕ)–
+ –v
∂ϕ
∂θ

∂u
∂v
+ (sinh ϕ)–
.
+ (uv)

∂ϕ
∂θ
(u – v )
From equation (.), we obtain D∗l ((f )– ) = . Therefore, the function (f )– is hyperholomorphic.
Lemma . Let be an open set in S. For p ∈ , if a split-quaternionic function f (x + Jy) =
u(x, y) + Jv(x, y) is hyperholomorphic, then f satisfies
D∗l f =
–v
.
r
(.)
Proof Since u(x, y) and v(x, y) are functions with respect to x and y, the calculation of D∗l f
is
 ∂u  ∂(Jv)
∂f  ∂f
∗
– J
–
–
Dl f =
∂t  ∂r
r ∂J r ∂J

∂(J)
∂(J)
=–
(Jθ )–
v + (Jϕ )–
v
r
∂θ
∂ϕ
v
=– .
r
Therefore, f satisfies equation (.).
Theorem . For p ∈ J , if a split-quaternionic function f (x + Jy) = u(x, y) + Jv(x, y) is
hyperholomorphic, then f satisfies
D∗l
f
rn
=
 ∗
f
D f – nJ n+ .
rn l
r
Proof Since u(x, y) and v(x, y) are functions with respect to x and y, the calculation of
D∗l ( rfn ) is
D∗l
f
rn
 ∂ f
 ∂ f
∂ f
=
– J
–
∂t rn
 ∂r rn
r ∂J rn
∂ f
∂v

  ∂u
=
J
r
–
un
+
J
r
–
vn
– n+ v
–
∂t rn
 rn+ ∂r
∂r
r

 ∂u  ∂v
 ∂v  ∂u
n u + Jv
= n
–
+J n
–
+ J n+ – n+ v
r ∂t  ∂r
r ∂t  ∂r
 r
r
=
n f

J
+ D∗ f .
 rn+ rn l
Therefore, we obtain the result.
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By referring to [, ] and observing Figures -, we consider the corresponding Gauss
theorem in four dimensions for the components of f in S. Specially, if the electric field is
known, the Gauss theorem can be of help to find the division of electric charge which is
inferred by integrating the electric field. For some symmetry, like cylindrical symmetry,
planar symmetry, and spherical symmetry, the electric field passes through the surfaces.
As shown in Figure , the Gauss theorem can be defined in the class containing the splitquaternionic variables with the cylindrical symmetry as a constituent. From Lemma .
and Theorem ., the Gauss theorem is described as follows.
Theorem . Let f = u(x, y) + J(x, y) be a hyperholomorphic function, and let K be any
smooth and simple closed hypersurface on T in S, disjoint from the real axis, K ∗ being the
interior of K . Let n(p) = n + n i + n j + n k, where (n , n , n , n ) is the unit outer normal
to K at p. Then

n
n(p)f (p) n dSK = –
r

K
K∗
J
u
rn+
dV ,
where dSk is the element of surface area on K .
Proof Suppose that f is hyperholomorphic and n(p) = n + n i + n j + n k, where
(n , n , n , n ) is the unit outer normal to K at p (such as Figure ). From Theorem .,
we have

n(p)f (p) n dSK =
r
K

(n f + in f + jn f + kn f ) dSK
n 
r
K
f

n f
dV
=
Dl n dV =
D
f
–
J
n l
r
 rn+
K∗
K∗ r
n u
–n –v
– J n+ + n
=
dV
 r
r r
K∗
n u
– J n+ dV .
=
 r
K∗
Thus, we can obtain the result.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This work was supported by the Dongguk University Research Fund of 2017.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 13 November 2016 Accepted: 6 March 2017
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