Integral theorems for monogenic functions in commutative algebras
Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We considermono...
Збережено в:
| Дата: | 2015 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2015
|
| Онлайн доступ: | https://trim.imath.kiev.ua/index.php/trim/article/view/308 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Transactions of Institute of Mathematics of NAS of Ukraine |
| Завантажити файл: |  |
Репозитарії
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552812893700096 |
|---|---|
| author | Shpakivskyi, V. S. Шпаківський, В. С. |
| author_facet | Shpakivskyi, V. S. Шпаківський, В. С. |
| author_institution_txt_mv | [
{
"author": "V. S. Shpakivskyi",
"institution": "Institute of Mathematics"
}
] |
| author_sort | Shpakivskyi, V. S. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-02T22:30:01Z |
| description | Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We considermonogenic (i.~e. continuous and differentiable in the sense of Gateaux) functions of the variable $\sum_{j=1}^k x_j\,e_j$  where $x_1,x_2,\ldots,x_k$ are real, and prove curvilinear analogues of the Cauchy integral theorem, the Morera theorem and the Cauchy integral formula in $k$-dimensional ($2\leq k\leq2n$)  real subspace of the algebra $\mathbb{A}_n^m$. The present  results are generalizations of the  corresponding results obtained in [1] for the case $k=3$. |
| first_indexed | 2026-08-04T01:05:23Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2015, Т.12, №4, 313–328
UDC 517.548
V. S. Shpakivskyi (Institute of Mathematics of the National Academy of
Sciences of Ukraine, Kyiv)
INTEGRAL THEOREMS FOR MONOGENIC FUNCTIONS
IN COMMUTATIVE ALGEBRAS
Let Am
n be an arbitrary n-dimensional commutative associative algebra over
the field of complex numbers with m idempotents. Let e1 = 1, e2, . . . , ek with
2 ≤ k ≤ 2n be elements of Am
n which are linearly independent over the field of
real numbers. We consider monogenic (i. e. continuous and differentiable in the
sense of Gateaux) functions of the variable
∑k
j=1 xj ej , where x1, x2, . . . , xk
are real, and prove curvilinear analogues of the Cauchy integral theorem, the
Morera theorem and the Cauchy integral formula in k-dimensional (2 ≤ k ≤
2n) real subspace of the algebra Am
n . The present results are generalizations of
the corresponding results obtained in [1] for the case k = 3.
1. Introduction. The Cauchy integral theorem and Cauchy integral
formula for the holomorphic function of the complex variable are
fundamental results of the classical complex analysis. Analogues of these
results are also important tools in commutative algebras of dimension
more than 2.
In the E. R. Lorch’s paper [2], for functions differentiable in the
sense of Lorch in an arbitrary convex domain of commutative associative
Banach algebra, some properties similar to properties of holomorphic
functions of complex variable (in particular, the curvilinear integral
Cauchy theorem and the integral Cauchy formula, the Taylor expansion
and the Morera theorem) are established. E. K. Blum [3] withdrew the
convexity of domain in the mentioned results from [2].
Let us note that a priori the differentiability of a function in the
sense of Gateaux is a restriction weaker than the differentiability of
this function in the sense of Lorch. Therefore, we consider a monogenic
functions defined as a continuous and differentiable in the sense of
Gateaux. Also we assume that a monogenic function is given in a
domain of three-dimensional subspace of an arbitrary commutative
c© V. S. Shpakivskyi, 2015
314 V. S. Shpakivskyi
associative algebra with unit over the field of complex numbers. In this
situation the results established in the papers [2, 3] is not applicable for
a mentioned monogenic function, because it deals with an integration
along a curve on which the function is not given, generally speaking.
In the papers [4–6] for monogenic function the curvilinear analogues
of the Cauchy integral theorem, the Cauchy integral formula and the
Morera theorem are obtained in special finite-dimensional commutative
associative algebras. The results of the papers [4, 5, 6] are generalized
in the paper [1] to an arbitrary commutative associative algebra. At the
same time, in [1] monogenic functions are given in a domain of a real
three-dimensional subspace of the algebra.
In this paper, assuming that monogenic functions are given in a
domain of a real k-dimensional subspace of the algebra, we generalize
results of the papers [1].
Let us note that some analogues of the curvilinear Cauchy’s integral
theorem and the Cauchy’s integral formula for other classes of functions
in special commutative algebras are established in the papers [7–10].
2. The algebra Amn . Let N be the set of natural numbers. We
fix numbers m,n ∈ N such that m ≤ n. Let Amn be an arbitrary
commutative associative algebra with unit over the field of complex
number C. E. Cartan [11, p. 33] proved that in the algebra Amn there
exist a basis {Ik}nk=1 which satisfies the following multiplication rules:
1. ∀ r, s ∈ [1,m] ∩ N : IrIs =
{
0 if r 6= s,
Ir if r = s;
2. ∀ r, s ∈ [m+ 1, n] ∩ N : IrIs =
n∑
k=max{r,s}+1
Υs
r,kIk ;
3. ∀ s ∈ [m+ 1, n] ∩ N ∃! us ∈ [1,m] ∩ N ∀ r ∈ [1,m] ∩ N :
IrIs =
{
0 if r 6= us ,
Is if r = us .
(1)
Furthermore, the structure constants Υs
r,k ∈ C satisfy the associativity
conditions:
(A 1). (IrIs)Ip = Ir(IsIp) ∀ r, s, p ∈ [m+ 1, n] ∩ N;
(A 2). (IuIs)Ip = Iu(IsIp) ∀ u ∈ [1,m]∩N ∀ s, p ∈ [m+ 1, n]∩N.
Integral theorems for monogenic functions in commutative algebras 315
Obviously, the first m basis vectors {Iu}mu=1 are idempotents and,
therefore, generate the semi-simple subalgebra. The vectors {Ir}nr=m+1
generate a nilpotent subalgebra of the algebra Amn . The unit of Amn is the
element of form 1 =
∑m
u=1 Iu. Therefore, we can state that the algebra
Amn is a semi-direct sum of the m-dimensional semi-simple subalgebra S
and (n−m)-dimensional nilpotent subalgebra N , i. e.
Amn = S ⊕s N. (2)
Let us note that nilpotent algebras are fully described for the
dimensions 1, 2, 3 in the paper [12], and some four-dimensional nilpotent
algebras can be found in the papers [13], [14].
The algebra Amn contains m maximal ideals
Iu :=
{
n∑
k=1, k 6=u
λkIk : λk ∈ C
}
, u = 1, 2, . . . ,m,
the intersection of which is the radical
R :=
{ n∑
k=m+1
λkIk : λk ∈ C
}
.
We define m linear functionals fu : Amn → C by putting
fu(Iu) = 1, fu(ω) = 0 ∀ω ∈ Iu , u = 1, 2, . . . ,m.
Since the kernel of every functional fu are the corresponding maximal
ideal Iu, these functionals are continuous and multiplicative (see [15,
p. 147]) also.
3. Monogenic functions in Ek. Consider vectors e1 = 1, e2, . . . , ek
in Amn , where 2 ≤ k ≤ 2n. Let these vectors be linearly independent over
the field of real numbers R (see [6]). It means that the equality
k∑
j=1
αjej = 0, αj ∈ R,
holds if and only if αj = 0 for all j = 1, 2, . . . , k.
316 V. S. Shpakivskyi
Let the vectors e1 = 1, e2, . . . , ek have the following decompositions
with respect to the basis {Ir}nr=1:
e1 =
m∑
r=1
Ir , ej =
n∑
r=1
ajr Ir , ajr ∈ C, j = 2, 3, . . . , k. (3)
Let ζ :=
k∑
j=1
xj ej , where xj ∈ R. It is obvious that
ξu := fu(ζ) = x1 +
k∑
j=2
xj aju, u = 1, 2, . . . ,m.
Let Ek := {ζ =
k∑
j=1
xjej : xj ∈ R} be the linear span of vectors
e1 = 1, e2, . . . , ek over the field R.
Everywhere below, we make the following essential assumption:
fu(Ek) = C for all u = 1, 2, . . . ,m. Obviously, it holds if and only if for
every fixed u = 1, 2, . . . ,m at least one of the numbers a2u, a3u, . . . , aku
belongs to C \ R.
With a set QR ⊂ Rk we associate the set Q := {ζ =
k∑
j=1
xjej :
(x1, . . . , xk) ∈ QR} in Ek. Note that topological properties of a set Q in
Ek are understood as corresponding topological properties of the set QR
in Rk. For example, the homotopy of a curve γ ⊂ Ek to the zero means
the homotopy of γR ⊂ Rk to the zero; the rectifiability of a curve γ ⊂ Ek
is understood as the rectifiability of the curve γR ⊂ Rk, etc.
Let Ω be a domain in Ek and
ΩR :=
{
(x1, x2, . . . , xk) ∈ Rk : ζ =
k∑
j=1
xj ej ∈ Ω
}
.
We say that a continuous function Φ : Ω→ Amn is monogenic in Ω if
Φ is differentiable in the sense of Gateaux in every point of Ω, i. e. if for
every ζ ∈ Ω there exists an element Φ′(ζ) ∈ Amn such that
lim
ε→0+0
(Φ(ζ + εh)− Φ(ζ)) ε−1 = hΦ′(ζ) ∀h ∈ Ek. (4)
Integral theorems for monogenic functions in commutative algebras 317
Φ′(ζ) is the Gateaux derivative of the function Φ at the point ζ.
Consider the decomposition of a function Φ : Ω → Amn with respect
to the basis {Ir}nr=1:
Φ(ζ) =
n∑
r=1
Ur(x1, x2, . . . , xk) Ir . (5)
In the case where the functions Ur : ΩR → C are R-differentiable in
ΩR, i. e. for every (x1, x2, . . . , xk) ∈ ΩR,
Ur (x1 + ∆x1, x2 + ∆x2, . . . , xk + ∆xk)− Ur(x1, x2, . . . , xk) =
=
k∑
j=1
∂Ur
∂xj
∆xj + o
√√√√ k∑
j=1
(∆xj)2
,
k∑
j=1
(∆xj)
2 → 0 ,
the function Φ is monogenic in the domain Ω if and only if the following
Cauchy – Riemann conditions are satisfied in Ω:
∂Φ
∂xj
=
∂Φ
∂x1
ej for all j = 2, 3, . . . , k. (6)
An expansion of the resolvent is of the form (see [16]):
(te1 − ζ)−1 =
m∑
u=1
1
t− ξu
Iu +
n∑
s=m+1
s−m+1∑
r=2
Qr,s
(t− ξus)
r Is (7)
∀ t ∈ C : t 6= ξu, u = 1, 2, . . . ,m,
where the coefficients Qr,s are determined by the following recurrence
relations:
Q2,s = Ts , Qr,s =
s−1∑
q=r+m−2
Qr−1,q Bq, s , r = 3, 4, . . . , s−m+ 1,
(8)
with
Ts :=
k∑
j=2
xjajs , Bq,s :=
s−1∑
p=m+1
TpΥ
p
q,s , p = m+ 2,m+ 3, . . . , n, (9)
318 V. S. Shpakivskyi
and the natural numbers us are defined in the rule 3 of the multiplication
table of algebra Amn .
In the paper [17] an expansion of the resolvent is obtained for the
case k = 3.
Consider the sets Mu := {ζ ∈ Ek : fu(ζ) = 0} for u = 1, 2, . . . ,m.
By Du ⊂ C we denote the image of Ω under the mapping fu, u =
1, 2, . . . ,m. We say that a domain Ω ⊂ Ek is convex with respect to the
set of directions Mu if Ω contains the segment {ζ1+α(ζ2−ζ1) : α ∈ [0, 1]}
for all ζ1, ζ2 ∈ Ω such that ζ2 − ζ1 ∈Mu.
In the next theorem we give a constructive description of all
monogenic functions given in domains of Ek and taking values in Amn
by means of holomorphic functions of the complex variable.
Theorem 1 [16]. Let a domain Ω ⊂ Ek be convex with respect to
the set of directions Mu and fu(Ek) = C for all u = 1, 2, . . . ,m. Then
every monogenic function Φ : Ω→ Amn can be expressed in the form
Φ(ζ) =
m∑
u=1
Iu
1
2πi
∫
Γu
Fu(t)(te1 − ζ)−1 dt+
+
n∑
s=m+1
Is
1
2πi
∫
Γus
Gs(t)(te1 − ζ)−1 dt, (10)
where Fu and Gs are certain holomorphic functions in the domains
Du and Dus , respectively, and Γq is a closed Jordan rectifiable curve
in Dq which surrounds the point ξq and contains no points ξ`, `, q =
1, 2, . . . ,m, ` 6= q.
Note that in the paper [17] the expression of the form (10) is proved
for the case k = 3.
4. Cauchy integral theorem for a curvilinear integral. Let γ
be a Jordan rectifiable curve in Ek. For a continuous function Ψ : γ →
Amn of the form
Ψ(ζ) =
n∑
r=1
Ur(x1, x2, . . . , xk) Ir + i
n∑
r=1
Vr(x1, x2, . . . , xk) Ir , (11)
Integral theorems for monogenic functions in commutative algebras 319
where (x1, x2, . . . , xk) ∈ γR and Ur : γR → R, Vr : γR → R, we define an
integral along a Jordan rectifiable curve γ by the equality:∫
γ
Ψ(ζ)dζ :=
k∑
j=1
ej
n∑
r=1
Ir
∫
γR
Ur(x1, x2, . . . , xk)dxj+
+i
k∑
j=1
ej
n∑
r=1
Ir
∫
γR
Vr(x1, x2, . . . , xk)dxj ,
where dζ := e1dx1 + e2dx2 + . . .+ ekdxk.
Let us define a surface integral also. Let Σ be a piece-smooth
hypersurface in Ek. For a continuous function Ψ : Σ → Amn of the form
(11), where (x1, x2, . . . , xk) ∈ ΣR and Ur : ΣR → R, Vr : ΣR → R, we
define a surface integral on Σ with the differential form dxp ∧ dxq, by
the equality∫
Σ
Ψ(ζ)dxp ∧ dxq :=
n∑
r=1
Ir
∫
ΣR
Ur(x1, x2, . . . , xk)dxp ∧ dxq+
+i
n∑
r=1
Ir
∫
ΣR
Vr(x1, x2, . . . , xk)dxp ∧ dxq.
If a function Φ : Ω → Amn is continuous together with partial
derivatives of the first order in a domain Ω, and Σ is a piece-smooth
hypersurface in Ω, and the edge γ of surface Σ is a rectifiable Jordan
curve, then the following analogue of the Stokes formula is true:∫
γ
Ψ(ζ)dζ =
∫
Σ
(
∂Ψ
∂x1
e2 −
∂Ψ
∂x2
e1
)
dx1 ∧ dx2+
+
(
∂Ψ
∂x2
e3 −
∂Ψ
∂x3
e2
)
dx2∧dx3+. . .+
(
∂Ψ
∂xk
e1 −
∂Ψ
∂x1
ek
)
dxk∧dx1. (12)
Now, the next theorem is a result of the formula (12) and the
equalities (6).
320 V. S. Shpakivskyi
Theorem 2. Suppose that Φ : Ω→ Amn is a monogenic function in
a domain Ω, and Σ is a piece-smooth surface in Ω, and the edge γ of
surface Σ is a rectifiable Jordan curve. Then∫
γ
Φ(ζ)dζ = 0. (13)
In the case where a domain Ω is convex, the equality (13) can be
proved for an arbitrary closed Jordan rectifiable curve γζ by the usual
way (see, e. g., [18]).
In the case where Ω is an arbitrary domain, similarly to the proof of
Theorem 3.2 [3], one can prove the following statement.
Theorem 3. Let Φ : Ω→ Amn be a monogenic function in a domain
Ω. Then for every closed Jordan rectifiable curve γ homotopic to a point
in Ω, the equality (13) is true.
5. The Morera theorem. To prove an analogue of Morera theorem
in the algebra Amn , we introduce auxiliary notions and prove some
auxiliary statements.
Let us consider the algebra Amn (R) with the basis {Ik, iIk}nk=1 over
the field R which is isomorphic to the algebra Amn over the field C. In
the algebra Amn (R) there exist another basis {er}2nr=1, where the vectors
e1, e2, . . . , ek are the same as in the Section 3.
For every element a :=
2n∑
r=1
arer, ar ∈ R, we define the Euclidian
norm
‖a‖ :=
√√√√ 2n∑
r=1
a2
r .
Accordingly, ‖ζ‖ =
√
k∑
j=1
x2
j and ‖ej‖ = 1 for all j = 1, 2, . . . , k.
Using the equivalence of norms in any finite-dimensional space, for
every element b :=
n∑
r=1
(b1r + ib2r)Ir with b1r, b2r ∈ R, we have the
Integral theorems for monogenic functions in commutative algebras 321
following inequalities:
|b1r + ib2r| ≤
√√√√ 2n∑
r=1
(
b21r + b22r
)
≤ c‖b‖, (14)
where c is a positive constant independent of b.
Lemma 1. If γ is a closed Jordan rectifiable curve in Ek and a
function Ψ : γ → Amn is continuous, then∥∥∥∥∥
∫
γ
Ψ(ζ) dζ
∥∥∥∥∥ ≤ c
∫
γ
‖Ψ(ζ)‖ ‖dζ‖, (15)
where c is a positive absolute constant.
Proof. Using the representation of function Ψ in the form (11) for
(x1, x2, . . . , xk) ∈ γ, we obtain
∥∥∥∥∥
∫
γ
Ψ(ζ)dζ
∥∥∥∥∥ ≤
n∑
r=1
‖e1Ir‖
∫
γR
∣∣Ur(x1, x2, . . . , xk)+iVr(x1, x2, . . . , xk)
∣∣ dx1+
. . .+
n∑
r=1
‖ekIr‖
∫
γR
∣∣Ur(x1, x2, . . . , xk) + i Vr(x1, x2, . . . , xk)
∣∣ dxk.
Now, taking into account the inequality (14) for b = Ψ(ζ) and the
inequalities ‖ejIr‖ ≤ cj , j = 1, 2, . . . , k, where cj are positive absolute
constants, we obtain the relation (15). The lemma is proved.
We understand a triangle 4 as a plane figure bounded by three line
segments connecting three its vertices. Denote by ∂4 the boundary of
triangle 4 in relative topology of its plane. We assume that the triangle
4 includes the boundary ∂4 also.
Using Lemma 1, for functions taking values in the algebra Amn , the
following Morera theorem can be established in the usual way.
Theorem 4. If a function Φ : Ω → Amn is continuous in a domain
322 V. S. Shpakivskyi
Ω and satisfies the equality ∫
∂4
Φ(ζ)dζ = 0 (16)
for every triangle 4 ⊂ Ω, then the function Φ is monogenic in the
domain Ω.
6. Cauchy integral formula for a curvilinear integral. Let
ζ0 :=
k∑
j=1
x
(0)
j ej be a point in a domain Ω ⊂ Ek. Let us take any 2-
dimensional plane containing the point ζ0. In this plane let us take a
circle C(ζ0, R) of radius R with the center at the point ζ0. Let R be such
that C(ζ0, R) is completely contained in Ω. By fu(C(ζ0, R)) we denote
the image of C(ζ0, R) under the mapping fu, u = 1, 2, . . . ,m.
We assume that the circle C(ζ0, R) embraces the set {ζ − ζ0 : ζ ∈
m⋃
u=1
Mu}. It means that the curve fu(C(ζ0, R)) bounds some domain D′u
and fu(ζ0) ∈ D′u for all u = 1, 2, . . . ,m.
We say that the curve γ ⊂ Ω embraces once the set {ζ − ζ0 : ζ ∈
m⋃
u=1
Mu}, if there exists a circle C(ζ0, R) which embraces the mentioned
set and is homotopic to γ in the domain Ω \ {ζ − ζ0 : ζ ∈
m⋃
u=1
Mu}.
Let a circle C(0, R) embrace the set
m⋃
u=1
Mu. Since the function ζ−1
is continuous on C(0, R), there exists the integral∫
C(0,R)
ζ−1dζ =: λ . (17)
The next theorem is an analogue of Cauchy integral theorem for
monogenic functions Φ : Ω→ Amn and is proved similarly to Theorem 4
in [1] .
Theorem 5. Suppose that a domain Ω ⊂ Ek is convex with respect to
the set of directions Mu and fu(Ek) = C for all u = 1, 2, . . . ,m. Suppose
Integral theorems for monogenic functions in commutative algebras 323
also that Φ : Ω → Amn is a monogenic function in Ω. Then for every
point ζ0 ∈ Ω the following equality is true:
λΦ(ζ0) =
∫
γ
Φ(ζ) (ζ − ζ0)
−1
dζ, (18)
where γ is an arbitrary closed Jordan rectifiable curve in Ω, that embraces
once the set {ζ − ζ0 : ζ ∈
m⋃
u=1
Mu}.
7. A constant λ. In certain special algebras (see [4–6]) the Cauchy
integral formula (18) has the form
Φ(ζ0) =
1
2πi
∫
γζ
Φ(ζ) (ζ − ζ0)
−1
dζ, (19)
i. e.
λ = 2πi. (20)
In this Section we indicate a set of algebras Amn for which (20) holds.
First, let us consider some auxiliary statements.
As a consequence of the expansion (7), we obtain the following
equality:
ζ−1 =
n∑
r=1
Ãr Ir (21)
with the coefficients Ãr determined by the following relations:
Ãu =
1
ξu
, u = 1, 2, . . . ,m,
Ãs =
s−m+1∑
r=2
Q̃r,s
ξrus
, s = m+ 1,m+ 2, . . . , n,
(22)
where Q̃r,s are determined by the following recurrence relations:
Q̃2,s = −Ts , Q̃r,s = −
s−1∑
q=r+m−2
Q̃r−1,q Bq, s , r = 3, 4, . . . , s−m+ 1,
(23)
324 V. S. Shpakivskyi
where Ts and Bq,s are the same as in the equalities (9), and natural
numbers us are defined in the rule 3 of the multiplication table of the
algebra Amn .
Taking into account the equality (21) and the relation
dζ =
k∑
j=1
dxj ej =
m∑
u=1
(
dx1 +
k∑
j=2
dxj aju
)
Iu+
+
n∑
r=m+1
k∑
j=2
dxj ajs Ir =
m∑
u=1
dξu Iu +
n∑
r=m+1
dTr Ir ,
we have the following equality:
ζ−1dζ =
m∑
u=1
Ãu dξu Iu +
n∑
r=m+1
Ãur dTr Ir+
+
n∑
s=m+1
Ãs dξus Is +
n∑
s=m+1
n∑
r=m+1
Ãs dTr IsIr =:
n∑
r=1
σr Ir . (24)
Now, taking into account the denotation (24) and the equalities (22),
we calculate:∫
C(0,R)
m∑
u=1
σu Iu =
m∑
u=1
Iu
∫
fu(C(0,R))
dξu
ξu
= 2πi
m∑
u=1
Iu = 2πi.
Thus,
λ = 2πi+
n∑
r=m+1
Ir
∫
C(0,R)
σr . (25)
Therefore, the equality (20) holds if and only if∫
C(0,R)
σr = 0 ∀ r = m+ 1, . . . , n. (26)
But, to satisfy the equality (26), the differential form σr must be the
total differential of some function. Note that the property to be the total
Integral theorems for monogenic functions in commutative algebras 325
differential is invariant under admissible transformations of coordinates
[19, p. 328, Theorem 2]. Thus, if to show that σr is the total differential
of a function which depends on the variables Tm+1
ξ , . . . , Tkξ , then it will
mean that σr is a total differential of a function of x1, x2, . . . , xk.
7.1. In this subsection we indicate a set of algebras in which the
vectors (3) are arbitrarily chosen and the equality (20) holds. In the
next theorem we use the representation (2) of algebra Amn .
Theorem 6. The equality (20) holds if at least one of the following
conditions are satisfied:
1) Amn ≡ S;
2) N is a zero nilpotent subalgebra;
3) dimCN ≤ 3, where dimCN is the complex dimensionality of
subalgebra N ;
4) dimCN = 4 and
Υm+1
m+1,m+2Υm+2
m+2,m+3 = Υm+1
m+1,m+2Υm+2
m+2,m+4 = Υm+1
m+1,m+3Υm+2
m+3,m+4 =
= Υm+3
m+3,m+4Υm+1
m+1,m+3 = Υm+1
m+2,m+3Υm+1
m+3,m+4 =
= Υm+1
m+2,m+3Υm+2
m+3,m+4 = Υm+1
m+2,m+3Υm+3
m+3,m+4 =
= Υm+1
m+1,m+2Υm+1
m+2,m+3Υm+2
m+3,m+4 = Υm+1
m+1,m+2Υm+1
m+2,m+3Υm+3
m+3,m+4 =
= Υm+2
m+2,m+3Υm+1
m+3,m+4 = Υm+2
m+2,m+3Υm+3
m+3,m+4 =
= Υm+2
m+2,m+3Υm+1
m+1,m+2Υm+1
m+3,m+4 = Υm+2
m+2,m+3Υm+1
m+1,m+2Υm+2
m+3,m+4 =
= Υm+2
m+2,m+3Υm+1
m+1,m+2Υm+3
m+3,m+4 = 0. (27)
The proof of Theorem 6 is analogous to the proofs of Theorems 5 —
8 in [1].
Further, we consider some examples of algebras which satisfy the
relations (27).
Examples.
• The algebra with the basis {I1 = 1, I2, I3, I4, I5} and the
multiplication rules
I2
2 = I3 , I2 I4 = I5,
326 V. S. Shpakivskyi
and other products of nilpotent elements I2, I3, I4, I5 are equal to
zero (for nilpotent subalgebra see [14], Table 21, algebra J69 and
[13], page 590, algebra A1,4).
• The algebra with the basis {I1 = 1, I2, I3, I4, I5} and the
multiplication rule
I2
2 = I3,
and other products of nilpotent elements I2, I3, I4, I5 are equal to
zero (for nilpotent subalgebra see [13], page 590, algebra A1,2 ⊕
A2
0,1).
• The algebra with the basis {I1 = 1, I2, I3, I4, I5} and the
multiplication rules
I2
2 = I3 , I
2
4 = I5,
and other products of nilpotent elements I2, I3, I4, I5 are equal to
zero (for nilpotent subalgebra see [13], page 590, algebra A1,2 ⊕
A1,2).
• The algebra with the basis {I1 = 1, I2, I3, I4, I5} and the
multiplication rules
I2
2 = I3 , I2 I3 = I4,
and other products of nilpotent elements I2, I3, I4, I5 are equal to
zero (for nilpotent subalgebra see [14], Table 21, algebra J71).
An example of algebra, which does not satisfy the relations (27), is
considered in the paper [1]. Moreover, in [1] the vectors e1, e2, e3 of form
(3) are selected such that the equality (20) is not true.
7.2 In this subsection we indicate sufficient conditions on a choose
of the vectors (3) for which the equality (20) is true. We use the
representation (2) of algebra Amn .
Theorem 7. If C(0, R) ⊂ Ek ⊂ S, then the equality (20) holds.
Proof. Note that the condition Ek ⊂ S means that ajr = 0 for all
j = 2, 3, . . . , k and r = m+ 1, . . . , n in the decomposition (3).
Integral theorems for monogenic functions in commutative algebras 327
If ζ ∈ S, then Ts = 0 for s = m + 1, . . . , n (see the denotation (9)).
It follows from the relation (24) that
σm+1 =
dTm+1
ξum+1
+ Ãm+1 dξum+1 ,
σr =
dTr
ξur
+ Ãr dξur +
r−1∑
q,s=m+1
Ãq dTsΥ
s
q,r , r = m+ 2, . . . , n.
(28)
Now, it follows from (23), (22) that Ãs = 0, and then σr = 0 for
r = m+ 1, . . . , n due to the equalities (28). Finally, the equality (20) is
a consequence of the equality σr = 0 and the relation (25). The theorem
is proved.
Theorem 7 generalizes Theorem 6 [20] and Theorem 9 [1].
Now, let us consider some results for the case where Ek 6⊂ S. First, if
dimCN ≤ 3, then by Theorem 6 the equality (20) holds for C(0, R) ⊂ Ek
and any Ek. Furthermore, using Theorem 6, we prove the next theorem
similarly to Theorem 10 in [1].
Theorem 8. Let dimCN = 4. Then the equality (20) holds if the
following two conditions are satisfied:
1) aj,m+1 = 0 for all j = 2, 3, . . . , k;
2) for every j = 2, 3, . . . , k at least one of the equalities aj,m+2 = 0
or aj,m+3 = 0 holds.
1. Shpakivskyi V. S. Curvilinear integral theorems for monogenic functions
in commutative associative algebras // Adv. Appl. Clifford Alg., DOI
10.1007/s00006-015-0561-x
2. Lorch E. R. The theory of analytic function in normed abelin vector rings //
Trans. Amer. Math. Soc., 54 (1943), 414 – 425.
3. Blum E. K. A theory of analytic functions in banach algebras // Trans. Amer.
Math. Soc., 78 (1955), 343 – 370.
4. Shpakivskyi V. S., Plaksa S. A. Integral theorems and a Cauchy formula in a
commutative three-dimensional harmonic algebra // Bulletin Soc. Sci. Lettr.
Lódź, 60 (2010), 47 – 54.
5. Plaksa S. A., Shpakivskyi V. S. Monogenic functions in a finite-dimensional
algebra with unit and radical of maximal dimensionality // J. Algerian Math.
Soc., 1 (2014), 1 – 13.
328 V. S. Shpakivskyi
6. Plaksa S. A., Pukhtaievych R. P. Constructive description of monogenic
functions in n-dimensional semi-simple algebra // An. Şt. Univ. Ovidius
Constanţa, 22 (2014), № 1, 221 – 235.
7. Ketchum P. W. Analytic functions of hypercomplex variables // Trans. Amer.
Math. Soc., 30 (1928), № 4, 641 – 667.
8. Ketchum P. W. A complete solution of Laplace’s equation by an infinite
hypervariable // Amer. J. Math., 51 (1929), 179 – 188.
9. Roşculeţ M. N. O teorie a funcţiilor de o variabilă hipercomplexă în spaţiul cu
trei dimensiuni // Studii şi Cercetǎri Matematice, 5, № 3–4 (1954), 361 – 401.
10. Roşculeţ M. N. Algebre liniare asociative şi comutative şi fincţii monogene
ataşate lor // Studii şi Cercetǎri Matematice, 6, № 1–2 (1955), 135 – 173.
11. Cartan E. Les groupes bilinéares et les systèmes de nombres complexes //
Annales de la faculté des sciences de Toulouse, 12 (1898), № 1, 1 – 64.
12. Burde D., de Graaf W. Classification of Novicov algebras // Applicable
Algebra in Engineering, Communication and Computing, 24(2013), no. 1, 1 –
15.
13. Burde D., Fialowski A. Jacobi–Jordan algebras // Linear Algebra Appl., 459
(2014), 586 – 594.
14. Martin M. E. Four-dimensional Jordan algebras // Int. J. Math. Game Theory
Algebra, 20 (4) (2013), 41 – 59.
15. Hille E., Phillips R. S. Functional analysis and semi-groups [Russian
translation], Inostr. Lit., Moscow (1962).
16. Shpakivskyi V. S. Monogenic functions in finite-dimensional commutative
associative algebras // accepted to Zb. Pr. Inst. Mat. NAN Ukr.
17. Shpakivskyi V. S. Constructive description of monogenic functions in a finite-
dimensional commutative associative algebra // submitted to Adv. Pure Appl.
Math., http://arxiv.org/pdf/1411.4643v1.pdf
18. Privalov I. I. Introduction to the Theory of Functions of a Complex Variable,
GITTL, Moscow, 1977. (Russian).
19. Shabat B. V. Introduction to complex analysis, Nauka, Moskow, 1961.
(Russian).
20. Plaksa S. A., Pukhtaevich R. P. Constructive description of monogenic
functions in a three-dimensional harmonic algebra with one-dimensional
radical // Ukr. Math. J., 65 (2013), № 5, 740 – 751.
|
| id | oai:trim.imath.kiev.ua:article-308 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:05:23Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/57/d77d926b632f9ccb1f630dc61ff51757.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3082018-02-02T22:30:01Z Integral theorems for monogenic functions in commutative algebras Інтегральні теореми для моногенних функцій в комутативних алгебах Shpakivskyi, V. S. Шпаківський, В. С. Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We considermonogenic (i.~e. continuous and differentiable in the sense of Gateaux) functions of the variable $\sum_{j=1}^k x_j\,e_j$  where $x_1,x_2,\ldots,x_k$ are real, and prove curvilinear analogues of the Cauchy integral theorem, the Morera theorem and the Cauchy integral formula in $k$-dimensional ($2\leq k\leq2n$)  real subspace of the algebra $\mathbb{A}_n^m$. The present  results are generalizations of the  corresponding results obtained in [1] for the case $k=3$. Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly independent over the field of real numbers. We considermonogenic (i.~e. continuous and differentiable in the sense of Gateaux) functions of the variable $\sum_{j=1}^k x_j\,e_j$  where $x_1,x_2,\ldots,x_k$ are real, and prove curvilinear analogues of the Cauchy integral theorem, the Morera theorem and the Cauchy integral formula in $k$-dimensional ($2\leq k\leq2n$)  real subspace of the algebra $\mathbb{A}_n^m$. The present  results are generalizations of the  corresponding results obtained in [1] for the case $k=3$. Інститут математики НАН України 2015-07-14 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/308 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 12 No. 4 (2015): Approximation Theory of Functions and Related Problems; 313 - 328 Сборник Трудов Института математики НАН Украины; Том 12 № 4 (2015): Tеорія наближення функцій та суміжні питання; 313 - 328 Збірник Праць Інституту математики НАН України; Том 12 № 4 (2015): Tеорія наближення функцій та суміжні питання; 313 - 328 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/308/338 Авторське право (c) 2015 V. V. Shpakivski |
| spellingShingle | Shpakivskyi, V. S. Шпаківський, В. С. Integral theorems for monogenic functions in commutative algebras |
| title | Integral theorems for monogenic functions in commutative algebras |
| title_alt | Інтегральні теореми для моногенних функцій в комутативних алгебах |
| title_full | Integral theorems for monogenic functions in commutative algebras |
| title_fullStr | Integral theorems for monogenic functions in commutative algebras |
| title_full_unstemmed | Integral theorems for monogenic functions in commutative algebras |
| title_short | Integral theorems for monogenic functions in commutative algebras |
| title_sort | integral theorems for monogenic functions in commutative algebras |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/308 |
| work_keys_str_mv | AT shpakivskyivs integraltheoremsformonogenicfunctionsincommutativealgebras AT špakívsʹkijvs integraltheoremsformonogenicfunctionsincommutativealgebras AT shpakivskyivs íntegralʹníteoremidlâmonogennihfunkcíjvkomutativnihalgebah AT špakívsʹkijvs íntegralʹníteoremidlâmonogennihfunkcíjvkomutativnihalgebah |