On the Cauchy theorem for hyperholomorphic functions of spatial variable
We proved a theorem about the integral of quaternionic-differentiable functions of spatial variable over the closed surface. It is an analog of the Cauchy theorem from complex analysis.
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Інститут прикладної математики і механіки НАН України
2017
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| Цитувати: | On the Cauchy theorem for hyperholomorphic functions of spatial variable / O.F. Herus // Український математичний вісник. — 2017. — Т. 14, № 2. — С. 153-160. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1693192020-06-11T01:26:28Z On the Cauchy theorem for hyperholomorphic functions of spatial variable Herus, O.F. We proved a theorem about the integral of quaternionic-differentiable functions of spatial variable over the closed surface. It is an analog of the Cauchy theorem from complex analysis. 2017 Article On the Cauchy theorem for hyperholomorphic functions of spatial variable / O.F. Herus // Український математичний вісник. — 2017. — Т. 14, № 2. — С. 153-160. — Бібліогр.: 15 назв. — англ. 1810-3200 2010 MSC. 30G35, 32V05 http://dspace.nbuv.gov.ua/handle/123456789/169319 en Український математичний вісник Інститут прикладної математики і механіки НАН України |
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We proved a theorem about the integral of quaternionic-differentiable functions of spatial variable over the closed surface. It is an analog of the Cauchy theorem from complex analysis. |
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Herus, O.F. |
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Herus, O.F. On the Cauchy theorem for hyperholomorphic functions of spatial variable Український математичний вісник |
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Herus, O.F. |
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Herus, O.F. |
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On the Cauchy theorem for hyperholomorphic functions of spatial variable |
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On the Cauchy theorem for hyperholomorphic functions of spatial variable |
| title_full |
On the Cauchy theorem for hyperholomorphic functions of spatial variable |
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On the Cauchy theorem for hyperholomorphic functions of spatial variable |
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On the Cauchy theorem for hyperholomorphic functions of spatial variable |
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on the cauchy theorem for hyperholomorphic functions of spatial variable |
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Інститут прикладної математики і механіки НАН України |
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2017 |
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On the Cauchy theorem for hyperholomorphic functions of spatial variable / O.F. Herus // Український математичний вісник. — 2017. — Т. 14, № 2. — С. 153-160. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний вісник |
| work_keys_str_mv |
AT herusof onthecauchytheoremforhyperholomorphicfunctionsofspatialvariable |
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2025-07-15T04:03:38Z |
| last_indexed |
2025-07-15T04:03:38Z |
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1837684195341631488 |
| fulltext |
Український математичний вiсник
Том 14 (2017), № 2, 153 – 160
On the Cauchy theorem for hyperholomorphic
functions of spatial variable
Oleg F. Herus
(Presented by V. Ya. Gutlyanskii)
Abstract. We proved a theorem about integral of quaternionic-
differentiable functions of spatial variable over the closed surface. It
is an analog of the Cauchy theorem from complex analysis.
2010 MSC. 30G35, 32V05.
Key words and phrases. Quaternion, Dirac operator, differentiable
function.
1. Introduction
Several researchers (see, e.g., [1, 2]) tried to generalize methods of
complex analysis onto analysis of functions acting in several-dimensional
algebras. At that, generalizations of different but mutually equivalent
definitions of holomorphy in complex analysis generate diverse classes of
hyperholomorphic functions in several-dimensional algebras.
Hypercomplex analysis in the space R3 was launched in the work of
G. Moisil and N. Theodoresco [3], where a three-dimensional analog of
the Cauchy–Riemann system was posed for the first time. R. Fueter [4]
first introduced a class of “regular” quaternion functions by means of
a four-dimensional generalization of the G. Moisil and N. Theodoresco
system. He proved quaternion analogues of the Cauchy theorem, the
integral Cauchy formula, the Liouville theorem and constructed an analog
of the Laurent series.
Now quaternion analysis gained wide evolution (more detailed see [1,
5–7]) previously thanks to its physical applications. At that in most works
it was usual to consider functions having continuous partial derivatives
in a domain and satisfying the above Cauchy–Riemann-type system. In
particular in the book [1] a spatial analog of the Cauchy theorem was
Received 20.03.2017
ISSN 1810 – 3200. c⃝ Iнститут прикладної математики i механiки НАН України
154 On the Cauchy theorem for hyperholomorphic...
proved by using the quaternion Stokes formula for bounded domains with
a piecewise-smooth boundary and for functions having continuous partial
derivatives in the closure of the domain.
In the survey paper [6] the continuity of partial derivatives was re-
placed by the weaker condition of real-differentiability for components
of the quaternion function. In the work [8] we consider the same class
of functions defined in a three-dimensional domain with the piecewise-
smooth boundary and requiring only the component-wise real-differenti-
ability and satisfying Cauchy–Riemann-type conditions like the class of
holomorphic functions in complex analysis (see e.g. [9]).
In the present work we extend the result of the paper [8] onto more
wide class of surfaces by using methods of the work [10], where a simi-
lar theorem was proved for functions taking values in finite-dimensional
commutative associative algebras.
2. Quaternion hyperholomorphic functions
Let H(C) be the associative algebra of complex quaternions
a =
3∑
k=0
akik,
where {ak}3k=0 ⊂ C, i0 = 1 and i1, i2, i3 be the imaginary quaternion
units with the multiplication rule i21 = i22 = i23 = i1i2i3 = −1. The
module of quaternion is defined by the formula
|a| :=
√√√√ 3∑
k=0
|ak|2.
Lemma 2.1 ( [11]). |ab| 6
√
2 |a| |b| for all {a; b} ⊂ H(C).
For {zk}3k=1 ⊂ R consider vector quaternions z := z1i1 + z2i2 + z3i3
as points of the Euclidean space R3 with the basis {ik}3k=1. Let Ω be
a domain of R3. For functions f : Ω → H(C) having first order partial
derivatives consider differential operators
Dl[f ] :=
3∑
k=1
ik
∂f
∂zk
,
Dr[f ] :=
3∑
k=1
∂f
∂zk
ik.
O. F. Herus 155
Definition 2.1. Function f := f0 + f1i1 + f2i2 + f3i3 is called left- or
right-H-differentiable in a point z(0) ∈ R3 if its components f0, f1, f2,
f3 are R3-differentiable functions in z(0) and the next condition
Dl[f ](z
(0)) = 0 (2.1)
or
Dr[f ](z
(0)) = 0
holds true respectively.
There is the notion of C-differentiability of a function f(ζ) = u(x, y)+
v(x, y)i, ζ = x+yi, in complex analysis (see [9, p. 33–34]). It is equivalent
to R2-differentiability in the point (x0, y0) of the components u(x, y) and
v(x, y) and satisfaction the condition
∂f(ζ0)
∂x
+
∂f(ζ0)
∂y
i = 0.
Thus the defined above notion of H-differentiability is the exact analog
of C-differentiability from complex analysis.
It is well known (see [9, p. 35]) that C-differentiability of a complex
function is equivalent to existence of its derivative. But in quaternion
analysis only the linear functions of special form have a derivative (see
[12]).
The operator Dl is called the Dirac operator (see [13]) or the Moisil–
Theodoresco operator (see [14]) and equality (2.1) is equivalent to the
Moisil–Theodoresco system [3].
Definition 2.2. A function f is called left- or right-hyperholomorphic
in a domain Ω if it is left- or right-H-differentiable in every point of the
domain.
3. Quaternion surface integral
Consider notions of surface and closed surface like to defined in the
work [10].
Definition 3.1. A surface Γ ⊂ R3 is an image of a closed set G ⊂ R2
under a homeomorphic mapping φ : G→ R3
φ(u, v) := (z1(u, v), z2(u, v), z3(u, v)), (u, v) ∈ G,
such that Jacobians
A :=
∂z2
∂u
∂z3
∂v
−∂z2
∂v
∂z3
∂u
, B :=
∂z3
∂u
∂z1
∂v
−∂z3
∂v
∂z1
∂u
, C :=
∂z1
∂u
∂z2
∂v
−∂z1
∂v
∂z2
∂u
exist almost everywhere on the set G and summable on G.
156 On the Cauchy theorem for hyperholomorphic...
The area of the surface Γ is calculated by the formula
L(Γ) =
∫∫
G
√
A2 +B2 + C2dudv,
where the integral is understood in the Lebesgue sense.
A surface Γ is called quadrable (see [10]) if L(Γ) < +∞.
Let Γ ⊂ R3 be an image of a sphere S ⊂ R3 in a such homeomorphic
mapping ψ : S → R3 that the image of a great circle γ on the sphere
S is a closed Jordan rectifiable curve γ̃ on the set Γ. The sphere S is
the union of two half-spheres S1, S2 with common edge γ. It is ease to
see that there exist continuously differentiable mappings φ1 : K → S1,
φ2 : K → S2 of the disk K := {(u, v) ∈ R2 : u2 + v2 6 1}. So the
set Γ is the union of two sets Γ1 = ψ(φ1(K)), Γ2 = ψ(φ2(K)) with the
intersection γ̃ = ψ(φ1(∂K)) = ψ(φ2(∂K)).
Definition 3.2. A set Γ is called a closed surface if there exist a such
homeomorphic mapping ψ : S → R3 that the sets Γ1, Γ2 are surfaces in
the sense of Definition 3.1 and orientation of the circle ∂K induces two
mutually opposite orientations of the curve γ̃ under mappings ψ ◦φ1 and
ψ ◦ φ2 respectively.
Let Γε :=
{
z ∈ R3 : ρ(z,Γ) 6 ε
}
(ρ denotes the Euclidean distance)
be the closed ε-neighborhood of the surface Γ, V (Γε) be the space Lebes-
gue measure of the set Γε and M∗(Γ) := lim
ε→0
V (Γε)
2ε be the two-dimensional
upper Minkowski content (see [15, p. 79]) of the surface Γ. For functions
f : Γ → H(C), g : Γ → H(C) in the case of non-closed quadrable surface
Γ the quaternion surface integral is defined by the formula∫∫
Γ
f(z)σ g(z) :=
∫∫
G
f(φ(u, v))(Ai1 +Bi2 + Ci3)g(φ(u, v))du dv,
where σ := dz2dz3i1 + dz3dz1i2 + dz1dz2i3, and in the case of a closed
surface — by the formula∫∫
Γ
f(z)σ g(z) :=
∫∫
Γ1
f(z)σ g(z) +
∫∫
Γ2
f(z)σ g(z).
In particular,
∫∫
Γ
|σ| = L(Γ).
Theorem 3.1 ( [8]). Let P be the surface of a closed cube contained in a
simply connected domain Ω ⊂ R3, let a function f : Ω → H(C) be right-
hyperholomorphic and a function g : Ω → H(C) be left-hyperholomorphic.
O. F. Herus 157
Then ∫∫
P
f(z)σ g(z) = 0.
Let δ > 0, let ωΓ(f, δ) := sup
|z1−z2|6δ
z1,z2 ∈Γ
|f(z1)− f(z2)| be the module of
continuity of a function f on Γ, and let d(Γ) be the diameter of Γ.
Lemma 3.1 ( [10]). Let Γ be a quadrable closed surface. Then∫∫
Γ
σ = 0. (3.1)
Lemma 3.2. Let Γ be a quadrable closed surface and let f : Ω → H(C)
and g : Ω → H(C) be continuous functions. Then∣∣∣∣∣∣
∫∫
Γ
f(z)σ g(z)
∣∣∣∣∣∣
6 2L(Γ)
(
ωΓ(f, d(Γ)) max
z∈Γ
|g(z)|+ ωΓ(g, d(Γ)) max
z∈Γ
|f(z)|
)
. (3.2)
Proof. Thanks to the formula (3.1) we have∫∫
Γ
f(z0)σ g(z0) = 0
for any point z0 ∈ Γ. Therefore∫∫
Γ
f(z)σ g(z) =
∫∫
Γ
(f(z)− f(z0))σ g(z0)
+
∫∫
Γ
f(z)σ (g(z)− g(z0)),
from which follows the estimate (3.2), taking into account Lemma 2.1.
Theorem 3.2. Let R3 ⊃ Ω be a bounded simply connected domain with
the quadrable closed boundary Γ, for which
M∗(Γ) < +∞, (3.3)
158 On the Cauchy theorem for hyperholomorphic...
let Ω have Jordan measurable intersections with planes perpendicular to
coordinate axes, let a function f : Ω → H(C) be right-hyperholomorphic
in Ω and continuous in the closure Ω and let a function g : Ω → H(C)
be left-hyperholomorphic in Ω and continuous in Ω. Then∫∫
Γ
f(z)σ g(z) = 0. (3.4)
Proof. Let us use the method proposed in the work [10] under the prov-
ing of Theorem 6.1. Thanks to the condition (3.3) there exists such a
constant c > 0 that for all sufficiently small ε > 0 the following inequality
holds
V (Γε) 6 cε. (3.5)
Decompose the space by planes perpendicular to coordinate axes onto
closed cubes with the edge of length ε√
3
. Let {Kj}, j ∈ J , be the finite
set of formed cubes having nonempty intersection with the surface Γ.
The integral (3.4) is representable in the form∫∫
Γ
f(z)σ g(z) =
∑
j∈J
∫∫
∂(Ω∩Kj)
f(z)σ g(z) +
∑
Kj⊂Ω
∫∫
∂Kj
f(z)σ g(z). (3.6)
By the Theorem 3.1 the second sum in the equality (3.6) is equal zero.
Every set Ω ∩ Kj consists of finite or infinite totality of connected
components. Applying the estimate (3.2) to the boundary of the every
component, we obtain∣∣∣∣∣∣∣
∫∫
∂(Ω∩Kj)
f(z)σ g(z)
∣∣∣∣∣∣∣
6 2(L(Γ ∩Kj) + 2ε2)
(
ωΓ(f, ε) max
z∈Ω
|g(z)|
+ ωΓ(g, ε) max
z∈Ω
|f(z)|
)
.
(3.7)
Substituting the inequality (3.7) into the equality (3.6), we obtain∣∣∣∣∣∣
∫∫
Γ
f(z)σ g(z)
∣∣∣∣∣∣
6 2
L(Γ) + 2
∑
j∈J
ε2
(ωΓ(f, ε) max
z∈Ω
|g(z)|+ ωΓ(g, ε) max
z∈Ω
|f(z)|
)
.
O. F. Herus 159
Since
∪
j∈J
Kj ⊂ Γε, we obtain from the inequality (3.5) that
1
3
√
3
∑
j∈J
ε3 6 V (Γε) 6 cε.
Therefore∣∣∣∣∣∣
∫∫
Γ
f(z)σ g(z)
∣∣∣∣∣∣
6 2(L(Γ) + 6
√
3c)
(
ωΓ(f, ε) max
z∈Ω
|g(z)|+ ωΓ(g, ε) max
z∈Ω
|f(z)|
)
and the equality (3.4) be obtained from here by passaging to the limit
when ε→ 0.
References
[1] V. V. Kravchenko, M. V. Shapiro, Integral representations for spatial models
of mathematical physics, Addison Wesley Longman, Pitman Research Notes in
Mathematics Series, 351, 1996.
[2] F. Brackx, R. Delanghe, F. Sommen, Clifford Analysis, Addison Wesley Longman.
Pitman Research Notes in Mathematics, 76, 1982.
[3] G. C. Moisil, N. Theodoresco, Functions holomorphes dans l’espace // Mathe-
matica (Cluj), 5 (1931), 142–159.
[4] R. Fueter, Die Funktionentheorie der Differentialgleichungen ∆u = 0 und ∆∆u =
0 mit vier reellen Variablen // Comment. Math. Helv., 8 (1936), 371–378.
[5] K. Gürlebeck, W. Sprössig, Quaternionic and Clifford Calculus for Physicists and
Engineers, John Wiley & Sons, 1997.
[6] A. Sudbery, Quaternionic Analysis // Math. Proc. Camb. Phil. Soc., 85 (1979),
199–225.
[7] V. V. Kravchenko, Applied quaternionic analysis, Heldermann-Verlag, Research
and Exposition in Mathematics Series, 28, 2003.
[8] O. F. Herus On hyperholomorphic functions of the space variable // Ukr. Mat.
Zh, 63, (2011), No. 4, 459–465.
[9] B. V. Shabat, Introduction to complex analysis. Part 1. Functions of one variable,
Moscow, Nauka, 1976 [in Russian].
[10] S. A. Plaksa, V. S. Shpakivskyi, Cauchy theorem for a surface integral in commu-
tative algebras // Complex Variables and Elliptic Equations, 59 (2014), No. 1,
110–119.
[11] O. F. Gerus, M. V. Shapiro, On a Cauchy-type integral related to the Helmholtz
operator in the plane // Boletin de la Sociedad Matemática Mexicana, 10 (2004),
N 1, 63–82.
[12] A. S. Meilikhzon, On the monogenity of quaternions // Doklady Akad. Nauk
SSSR, 59 (1948), No. 3, 431–434.
160 On the Cauchy theorem for hyperholomorphic...
[13] J. Cnops, An introduction to Dirac operators on manifolds, Progress in Mathe-
matical Physics. 24. Birkhäuser, Boston, 2002.
[14] R. A. Blaya, J. B. Reyes, M. Shapiro, On the Laplasian vector fields theory in
domains with rectifiable boundary // Mathematical Methods in the Applied Sci-
ences, 29 (2006), 1861–1881.
[15] P. Mattila, Geometry of sets and measures in Euclidean spaces. Fractals and
rectifiability, Cambridge, Cambridge University Press, 1995.
Contact information
Oleg F. Herus Zhytomyr Ivan Franko State University,
Zhytomyr, Ukraine
E-Mail: ogerus@zu.edu.ua
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