Monogenic functions in finite-dimensional commutative associative algebras
Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra overthe field of complex numbers with $m$ idempotents. Let $e_1=1$,\break $e_2,\ldots,e_k$, $2\leqk\leq 2n$, are linearly independent over the field of real numbers elements of $\mathbb{A}_n^m$.We consider monogenic...
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| author | Shpakivskyi, V. S. Shpakivskyi, V. S. |
| author_facet | Shpakivskyi, V. S. Shpakivskyi, V. S. |
| author_institution_txt_mv | [
{
"author": "V. S. Shpakivskyi",
"institution": "Institute of Mathematics of NAS of Ukraine"
}
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| description | Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra overthe field of complex numbers with $m$ idempotents. Let $e_1=1$,\break $e_2,\ldots,e_k$, $2\leqk\leq 2n$, are linearly independent over the field of real numbers elements of $\mathbb{A}_n^m$.We consider monogenic (i.~e., continuous and differentiable in the sense of Gateaux) functions ofthe variable $\sum_{j=1}^k x_j\,e_j$\,, where $x_1,x_2,\ldots,x_k$ are real, and obtain aconstructive description of all mentioned functions by means of holomorphic functions of complexvariables. Due to this description obtain, that monogenic functions have Gateaux derivatives ofall orders. The present article is a generalization of the author's paper \cite{Shpakivskyi-2014},where mentioned results are obtained for $k=3$. |
| first_indexed | 2026-08-04T01:03:33Z |
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Збiрник праць Iн-ту математики НАН України 2015, том 12, № 3, 251–268
УДК 517.548
V. S. Shpakivskyi
(Institute of Mathematics of NAS of Ukraine, Kyiv)
shpakivskyi86@gmail.com, shpakivskyi@imath.kiev.ua
Monogenic functions in
finite-dimensional commutative
associative 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, 2 ≤ k ≤ 2n, are linearly independent over the field of real
numbers elements of Am
n . 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 obtain a constructive description of
all mentioned functions by means of holomorphic functions of complex
variables. Due to this description obtain, that monogenic functions have
Gateaux derivatives of all orders. The present article is a generalization of
the author’s paper [1], where mentioned results are obtained for k = 3.
1. Introduction. It seemed, W. Hamilton (1843) made the first
attempts to construct an algebra associated with the three-dimensional
Laplace equation
∆3u(x, y, z) :=
(
∂2
∂x2 +
∂2
∂y2
+
∂2
∂z2
)
u(x, y, z) = 0 (1)
meaning that components of hypercomplex functions satisfy the Eq. (1).
He constructed an algebra of noncommutative quaternions over the field of
real numbers R and made a base for developing the hypercomplex analysis.
C. Segre [2] constructed an algebra of commutative quaternions over
the field R that can be considered as a two-dimensional commutative semi-
simple algebra of bicomplex numbers over the field of complex numbers
c⃝ Institute of Mathematics, 2015
252 V. S. Shpakivskyi
C. M. Futagawa [3] and J. Riley [4] obtained a constructive description of
analytic function of a bicomplex variable, namely, they proved that such an
analytic function can be constructed with use of two holomorphic functions
of complex variables.
F. Ringleb [5] and S. N. Volovel’skaya [6, 7] succeeded in developing a
function theory for noncommutative algebras with unit over the real or
complex fields, by pursuing a definition of the differential of a function
on such an algebra suggested by Hausdorff in [8]. These definitions make
the a priori severe requirement that the coordinates of the function
have continuous first derivatives with respect to the coordinates of the
argument element. Namely, F. Ringleb [5] considered an arbitrary finite-
dimensional associative (commutative or not) semi-simple algebra over
R. For given class of functions which maps the mentioned algebra onto
itself, he obtained a constructive description by means of real and complex
analytic functions.
S. N. Volovel’skaya developed the Hausdorff’s idea defining the
monogenic functions on non-semisimple associative algebras and she
generalized the Ringleb’s results for such algebras. In [6], there was
obtained a constructive description of monogenic functions in a special
three-dimensional non-commutative algebra over the field R. The results
of paper [6] were generalized in the paper [7], where Volovel’skaya obtained
a constructive description of monogenic functions in non-semisimple
associative algebras of the first category over R.
A relation between spatial potential fields and analytic functions given
in commutative algebras was established by P. W. Ketchum [9], who shown
that every analytic function Φ(ζ) of the variable ζ = xe1+ye2+ze3 satisfies
the Eq. (1) in the case where the elements e1, e2, e3 of a commutative
algebra satisfy the condition
e21 + e22 + e23 = 0 , (2)
because
∂2Φ
∂x2 +
∂2Φ
∂y2
+
∂2Φ
∂z2
≡ Φ′′(ζ) (e21 + e22 + e23) = 0 , (3)
where Φ′′ := (Φ′)′, Φ′(ζ) is defined by the equality dΦ = Φ′(ζ)dζ.
We say that a commutative associative algebra A is harmonic (cf. [9–
11]) if in A there exists a triad of linearly independent vectors {e1, e2, e3}
satisfying the equality (2) with e2k ̸= 0 for k = 1, 2, 3. We say also that
such a triad {e1, e2, e3} is harmonic.
Monogenic functions in finite-dimensional commutative ... 253
P. W. Ketchum [9] considered the C. Segre algebra of quaternions [2]
as an example of a harmonic algebra.
Further M. N. Roşculeţ establishes a relation between monogenic
functions in commutative algebras and partial differential equations. He
defined monogenic functions f of the variable w by the equality df(w) dw =
0. In [12], M. N. Roşculeţ proposed a procedure for constructing an infinite-
dimensional topological vector space with commutative multiplication such
that monogenic functions in it are the all solutions of the equation∑
α0+α1+...+αp=N
Cα0,α1,...,αp
∂NΦ
∂xα0
0 ∂xα1
1 . . . ∂x
αp
p
= 0, (4)
where Cα0,α1,...,αp ∈ R. In particular, such infinite-dimensional topological
vector space are constructed for the Laplace equation (3). In [13], Roşculeţ
finds a certain connection between monogenic functions in commutative
algebras and systems of partial differential equations.
I. P. Mel’nichenko proposed to use hypercomplex functions
differentiable in the sense of Gateaux for describing solutions of the
equation (4), since conditions of monogenicity are the least restrictive
in this case. He started to implement this approach with respect to
the 3-D Laplace equation (3) (see [10]). Mel’nichenko proved that exist
exactly 3 three-dimensional harmonic algebras with unit over the field C
(see [10,11,14]).
In [15], it is developed the Melnichenko’s idea for the equation (4).
An investigation of partial differential equations using the
hypercomplex methods is more effective if hypercomplex monogenic
(in any sense) functions can be constructed explicitly. Constructive
descriptions of monogenic (i. e. continuous and differentiable in the sense
of Gateaux) functions taking values in the mentioned three-dimensional
harmonic algebras by means of three corresponding holomorphic functions
of the complex variable are obtained in [16–18]. Such descriptions make
it possible to prove the infinite differentiability (in the sense of Gateaux)
of monogenic functions and integral theorems for these functions, being
analogous to classical theorems in Complex Analysis (see, e. g., [19, 20]).
Furthermore, constructive descriptions of monogenic functions taking
values in special n-dimensional commutative algebras by means n
holomorphic functions of complex variables are obtained in [21,22].
In [1], there is obtained a constructive description of all monogenic
functions of the variable x1e1 + x2e2 + x3e3 taking values in an arbitrary
254 V. S. Shpakivskyi
n-dimensional commutative associative algebra with unit by means of
holomorphic functions of the complex variables. It follows from this
description that monogenic functions are infinitely differentiable in the
sense of Gateaux.
In this paper we extend results of the paper [1] to monogenic functions
of the variable
k∑
r=1
xrer, where 2 ≤ r ≤ 2n.
2. The algebra Am
n . Let N be the set of natural numbers. We fix
ordered numbers m,n ∈ N, m ≤ n. Let Am
n be an arbitrary commutative
associative algebra with unit over the field of complex number C. E. Cartan
[23, p. 33] proved that there exists a basis {Ir}nr=1 in Am
n satisfying the
following multiplication rules:
1. ∀ r, s ∈ [1,m] ∩ N : IrIs =
{
0 if r ̸= s,
Ir if r = s;
2. ∀ r, s ∈ [m+ 1, n] ∩ N : IrIs =
n∑
p=max{r,s}+1
Υs
r,pIp ;
3. ∀ s ∈ [m+ 1, n] ∩ N ∃! us ∈ [1,m] ∩ N ∀ r ∈ [1,m] ∩ N :
IrIs =
{
0 if r ̸= us ,
Is if r = us .
(5)
Moreover, the structure constants Υs
r,p ∈ 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.
Obviously, that the first m basic vectors {Iu}mu=1 are idempotents and
define the basis of the semi-simple subalgebra of the algebra Am
n . The
vectors {Ir}nr=m+1 define the basis of the nilpotent subalgebra of the
algebra Am
n . The element 1 =
∑m
u=1 Iu is the unit of Am
n .
In the cases where
Consider some particular cases of Am
n .
Proposition 1 [1]. If there exists the unique u0 ∈ [1,m]∩N such that
Iu0Is = Is for all s = m + 1, . . . , n, then the associativity condition (A 2)
Monogenic functions in finite-dimensional commutative ... 255
is satisfied.
Thus, under the conditions of Proposition 1, the associativity condition
(A 1) is merely required. It means that the nilpotent subalgebra of Am
n
with the basis {Ir}nr=m+1 can be an arbitrary commutative associative
nilpotent algebra of dimension n −m. Note, that such nilpotent algebras
are completely described for the dimensions 1, 2, 3 in the paper [24], and
some four-dimensional nilpotent algebras can be found in [25,26].
Proposition 2 [1]. If all ur are distinct in the multiplication rule 3,
then IsIp = 0 for all s, p = m+ 1, . . . , n.
Thus, under the conditions of Proposition 2, the multiplication table
of the nilpotent subalgebra of Am
n with the basis {Ir}nr=m+1 consists only
of zeros and all associativity conditions are satisfied.
The algebra Am
n contains m maximal ideals
Iu :=
{
n∑
r=1, r ̸=u
λrIr : λr ∈ C
}
, u = 1, 2, . . . ,m,
and their intersection is the radical
R :=
{ n∑
r=m+1
λrIr : λr ∈ C
}
.
Consider m linear functionals fu : Am
n −→ C satisfying the equalities
fu(Iu) = 1, fu(ω) = 0 ∀ω ∈ Iu , u = 1, 2, . . . ,m.
Inasmuch as the kernel of functional fu is the maximal ideal Iu obtain,
that this functional is also continuous and multiplicative (see [27, p. 147]).
3. Monogenic functions. Let us consider linearly
independent over the field of real numbers R (see [22]) vectors
e1 = 1, e2, . . . , ek in Am
n , where 2 ≤ k ≤ 2n. 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.
Let the vectors {e1, . . . , ek} have the following decompositions in the
basis {Ir}nr=1:
e1 =
m∑
r=1
Ir , ej =
n∑
r=1
ajr Ir , ajr ∈ C, j = 2, 3, . . . , k. (6)
256 V. S. Shpakivskyi
Let ζ :=
k∑
j=1
xj ej , xj ∈ R. Evidently, 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, . . . , ek} over R.
Let Ω be a domain in Ek. With a domain Ω ⊂ Ek we associate the
domain ΩR :=
{
(x1, x2, . . . , xk) ∈ Rk : ζ =
k∑
j=1
xj ej ∈ Ω
}
.
We say that a continuous function Φ: Ω −→ Am
n is monogenic in Ω if Φ
is differentiable in the sense of Gateaux in Ω, i.e., there exists an element
Φ′(ζ) ∈ Am
n such that
lim
ε→0+0
(Φ(ζ + εh)− Φ(ζ)) ε−1 = hΦ′(ζ) ∀h ∈ Ek. (7)
in any ζ ∈ Ω. The function Φ′(ζ) is the Gateaux derivative of the function
Φ in the point ζ.
Consider the decomposition of a function Φ: Ω −→ Am
n in the basis
{Ir}nr=1:
Φ(ζ) =
n∑
r=1
Ur(x1, x2, . . . , xk) Ir . (8)
If functions Ur : ΩR −→ C are R-differentiable in ΩR, i.e.,
for every (x1, x2, . . . , xk) ∈ ΩR the following asymptotic equality
is valid: 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. (9)
4. Expansion of the resolvent. Let b :=
n∑
r=1
br Ir ∈ Am
n , where
br ∈ C. Note, that fu(b) = bu, u = 1, 2, . . . ,m. It follows form Lemmas 1,
Monogenic functions in finite-dimensional commutative ... 257
3 in [1] that
b−1 =
m∑
u=1
1
bu
Iu +
n∑
s=m+1
s−m+1∑
r=2
Q̃r,s
brus
Is, (10)
where Q̃r,s are determined by the following recurrence relations:
Q̃2,s := bs , Q̃r,s =
s−1∑
q=r+m−2
Q̃r−1,q B̃q, s , r = 3, 4, . . . , s−m+ 1,
(11)
B̃q,s :=
s−1∑
p=m+1
bpΥ
p
q,s , p = m+ 2,m+ 3, . . . , n, (12)
and the natural numbers us are defined by the rule 3 of the multiplication
table of the algebra Am
n .
In the next lemma we find an expansion of the resolvent (te1 − ζ)−1.
Lemma 1. The resolvent has the following expansion
(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 (13)
∀ t ∈ C : t ̸= ξu, u = 1, 2, . . . ,m,
where 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,
(14)
here
Ts :=
k∑
j=2
xjajs , Bq,s :=
s−1∑
p=m+1
TpΥ
p
q,s , p = m+ 2,m+ 3, . . . , n, (15)
and the natural numbers us are defined by the rule 3 of the multiplication
table of the algebra Am
n .
Proof. Taking into account the decomposition te1 − ζ =
=
m∑
u=1
(t − ξu)Iu −
n∑
r=m+1
k∑
j=2
xjajs Ir , conclude, that the relation (13)
258 V. S. Shpakivskyi
follows directly from the equality (10) in which instead of bu, u =
= 1, 2, . . . ,m, it should be used the expansion t − ξu; and instead of bs ,
s = m+ 1,m+ 2, . . . , n, it should be used the expansion
k∑
j=2
xjajs. The
lemma is proved.
It follows from Lemma 1 that points (x1, x2, . . . , xk) ∈ Rk, which are
correspond to the non-invertible elements ζ =
k∑
j=1
xj ej , form the set
MR
u :
x1 +
k∑
j=2
xj Re aju = 0,
k∑
j=2
xj Im aju = 0,
u = 1, 2, . . . ,m
in the k-dimensional space Rk. Consider the set Mu := {ζ ∈ Ek :
: fu(ζ) = 0} for u = 1, 2, . . . ,m. It is obvious that the set MR
u ⊂ Rk
is congruent to the set Mu ⊂ Ek.
5. A constructive description of monogenic functions.
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.
Denote fu(Ek) := {fu(ζ) : ζ ∈ Ek}. In what follows, 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.
Further in this section, we 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.
Lemma 2. Suppose that a function Φ: Ω −→ Am
n is monogenic in
the domain Ω. If points ζ1, ζ2 ∈ Ω such that ζ2 − ζ1 ∈ Mu, then
Φ(ζ2)− Φ(ζ1) ∈ Iu . (16)
Proof. Inasmuch as fu(Ek) = C then exists an element e∗2 ∈ Ek such
that fu(e
∗
2) = i. Consider the lineal span E∗ := {ζ = xe∗1 + ye∗2 + ze∗3 :
x, y, z ∈ R} of the vectors e∗1 := 1, e∗2, e
∗
3 := ζ2 − ζ1.
Monogenic functions in finite-dimensional commutative ... 259
Now, the relations (16) can be proved in such a way as Lemma 2.1 [16],
in the proof of which one must take Ω ∩ E∗, fu, {αe∗3 : α ∈ R} instead of
Ωζ , f, L, respectively. Lemma 2 is proved.
Let a domain Ω ⊂ Ek be convex with respect to the set of directions
Mu , u = 1, 2, . . . ,m, Du := fu(Ω) ⊂ C.
We introduce linear operators Au , u = 1, 2, . . . ,m, which assign
holomorphic functions Fu : Du −→ C to monogenic functions Φ: Ω −→ Am
n
by the formula
Fu(ξu) = fu(Φ(ζ)), (17)
where ξu = fu(ζ) ≡ x1 +
k∑
j=2
xj aju and ζ ∈ Ω. It follows from Lemma 2
that the value Fu(ξu) does not depend on a choice of a point ζ for which
fu(ζ) = ξu.
Now, similar to proof of Lemma 5 [1] it can be proved the following
statement.
Lemma 3. Suppose that for any fixed u = 1, 2, . . . ,m, a function
Fu : Du −→ C is holomorphic in a domain Du and Γu is a closed Jordan
rectifiable curve in Du which surrounds the point ξu and contains no points
ξq, q = 1, 2, . . . ,m, q ̸= u. Then the function
Ψu(ζ) := Iu
∫
Γu
Fu(t)(te1 − ζ)−1 dt (18)
is monogenic in the domain Ω.
Lemma 4. Suppose that a function V : ΩR −→ C satisfies the
equalities
∂V
∂x2
=
∂V
∂x1
a2u ,
∂V
∂x3
=
∂V
∂x1
a3u , . . . ,
∂V
∂xk
=
∂V
∂x1
aku (19)
in ΩR. Then V is a holomorphic function of the variable ξu = fu(ζ) =
= x1 +
k∑
j=2
xj aju in the domain Du.
Proof. We first separate the real and the imaginary part of the
expression
ξu = x1 +
k∑
j=2
xj Re aju + i
k∑
j=2
xj Im aju =: τu + iηu (20)
260 V. S. Shpakivskyi
and note that the equalities (19) yield
∂V
∂ηu
Im a2u = i
∂V
∂τu
Im a2u , . . . ,
∂V
∂ηu
Im aku = i
∂V
∂τu
Im aku . (21)
It follows from the condition fu(Ek) = C that at least one of the
numbers Im a2u , Im a3u , . . . , Im bu is not equal to zero. Therefore, using
(21), we get
∂V
∂ηu
= i
∂V
∂τu
. (22)
Now we prove that V (x′
1, x
′
2, . . . , x
′
k) = V (x′′
1 , x
′′
2 , . . . , x
′′
k) for points
(x′
1, x
′
2, . . . , x
′
k), (x
′′
1 , x
′′
2 , . . . , x
′′
k) ∈ Ω such that the segment connecting
these points is parallel to a straight line Lu ⊂ MR
u . We use considerations
with the proof of Lemma 2. Since fu(Ek) = C, then there exists an
element e∗2 ∈ Ek such that fu(e
∗
2) = i. Consider the lineal span E∗ :=
:= {ζ = xe∗1+ye∗2+ze∗3 : x, y, z ∈ R} of the vectors e∗1 := 1, e∗2, e∗3 := ζ ′−ζ ′′,
where ζ ′ :=
k∑
j=1
x′
j ej , ζ ′′ :=
k∑
j=1
x′′
j ej .
Now, the relation V (x′
1, x
′
2, . . . , x
′
k) = V (x′′
1 , x
′′
2 , . . . , x
′′
k) can be proved
in such a way as Lemma 6 [1], in the proof of which one must take
Ω∩E∗,{αe∗3 : α ∈ R} instead of Ωζ , L, respectively. The lemma is proved.
Thus, a function V : ΩR −→ C of the type V (x1, x2, . . . , xk) := F (ξu),
where F (ξu) is an arbitrary function holomorphic in the domain Du , is a
general solution of the system (19). The lemma is proved.
Theorem 1. Every monogenic function Φ : Ω → Am
n 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, (23)
where Fu and Gs are certain holomorphic functions in the domains Du
and Dus
, respectively; Γq is a closed Jordan rectifiable curve in Dq which
surrounds the point ξq and contains no points ξℓ, ℓ, q = 1, 2, . . . ,m, ℓ ̸= q.
Proof. We set
Fu := AuΦ, u = 1, 2, . . . ,m. (24)
Monogenic functions in finite-dimensional commutative ... 261
Let us show that the values of monogenic function
Φ0(ζ) := Φ(ζ)−
m∑
u=1
Iu
1
2πi
∫
Γu
Fu(t)(te1 − ζ)−1 dt (25)
belong to the radical R, i.e., Φ0(ζ) ∈ R for all ζ ∈ Ω. As a consequence of
the equality (13), we have the equality
Iu
1
2πi
∫
Γu
Fu(t)(te1 − ζ)−1 dt = Iu
1
2πi
∫
Γu
Fu(t)
t− ξu
dt+
+
1
2πi
n∑
s=m+1
s−m+1∑
r=2
∫
Γu
Fu(t)Qr,s
(t− ξus)
r dt Is Iu ,
from which we obtain the equality
fu
( m∑
u=1
Iu
1
2πi
∫
Γu
Fu(t)(te1 − ζ)−1 dt
)
= Fu(ξu). (26)
Acting the functional fu onto the equality (25) and taking into
account the relations (17), (24), (26), we get the equality fu(Φ0(ζ)) =
= Fu(ξu)− Fu(ξu) = 0 for all u = 1, 2, . . . ,m, i.e., Φ0(ζ) ∈ R.
Therefore, the function Φ0 is of the type
Φ0(ζ) =
n∑
s=m+1
Vs(x1, x2, . . . , xk) Is , (27)
where functions Vs, s = m + 1, . . . , n, are of the type Vs : ΩR −→ C.
Cauchy–Riemann conditions (9) are satisfied with Φ = Φ0. Substituting
the expressions (6), (27) into the equality (9), we obtain
n∑
s=m+1
∂Vs
∂x2
Is =
n∑
s=m+1
∂Vs
∂x1
Is
n∑
r=1
a2r Ir ,
...
n∑
s=m+1
∂Vs
∂xk
Is =
n∑
s=m+1
∂Vs
∂x1
Is
n∑
r=1
akr Ir .
(28)
262 V. S. Shpakivskyi
Equating the coefficients near Im+1 in these equalities, we obtain the
following system of equations with unknown function Vm+1(x1, x2, . . . , xk):
∂Vm+1
∂x2
=
∂Vm+1
∂x1
a2um+1 , . . . ,
∂Vm+1
∂xk
=
∂Vm+1
∂x1
ak um+1 .
It follows from Lemma 4 that Vm+1(x1, x2, . . . , xk) ≡ Gm+1(ξum+1), where
Gm+1 is a function holomorphic in the domain Dum+1 . Therefore,
Φ0(ζ) = Gm+1(ξum+1) Im+1 +
n∑
s=m+2
Vs(x1, x2, . . . , xk) Is . (29)
Due to the expansion (13), we have the representation
Im+1
1
2πi
∫
Γum+1
Gm+1(t)(te1 − ζ)−1 dt = Gm+1(ξum+1) Im+1 +Ψ(ζ), (30)
where Ψ(ζ) is a function with values in the set
{∑n
s=m+2 αs Is : αs ∈ C
}
.
Now, consider the function
Φ1(ζ) := Φ0(ζ)− Im+1
1
2πi
∫
Γum+1
Gm+1(t)(te1 − ζ)−1 dt.
In view of the relations (29), (30), Φ1 can be represented in the form
Φ1(ζ) =
n∑
s=m+2
Ṽs(x1, x2, . . . , xk) Is ,
where functions Ṽs, s = m+ 2, . . . , n, are of the type Ṽs : ΩR −→ C .
Inasmuch as Φ1 is a monogenic function in Ω, the functions
Ṽm+2, Ṽm+3, . . . , Ṽn satisfy the system (28) with Vm+1 ≡ 0,
Vs = Ṽs for s = m + 2,m + 3, . . . , n. Therefore, similarly to the function
Vm+1(x1, x2, . . . , xk) ≡ Gm+1(ξum+1), the function Ṽm+2 satisfies the
equations
∂Ṽm+2
∂x2
=
∂Ṽm+2
∂x1
a2um+2 , . . . ,
∂Ṽm+2
∂xk
=
∂Ṽm+2
∂x1
ak um+2
and is of the form Ṽm+2(x1, x2, . . . , xk) ≡ Gm+2(ξum+2
), where Gm+2 is a
function holomorphic in the domain Dum+2 .
Monogenic functions in finite-dimensional commutative ... 263
In such a way, step by step, considering the functions
Φj(ζ) := Φj−1(ζ)− Im+j
1
2πi
∫
Γum+j
Gm+j(t)(te1 − ζ)−1 dt
for j = 2, 3, . . . , n−m− 1, we get the representation (23) of the function
Φ. The theorem is proved.
Taking into account the expansion (13), one can rewrite the equality
(23) in the following equivalent form:
Φ(ζ) =
m∑
u=1
Fu(ξu)Iu +
n∑
s=m+1
s−m+1∑
r=2
1
(r − 1)!
Qr,s F
(r−1)
us
(ξus
) Is+
+
n∑
q=m+1
Gq(ξuq )Iq +
n∑
q=m+1
n∑
s=m+1
s−m+1∑
r=2
1
(r − 1)!
Qr,s G
(r−1)
q (ξuq ) Iq Is .
(31)
Thus, the equalities (23) and (31) rebuild any monogenic functions
Φ : Ω → Am
n by n corresponding holomorphic functions of the complex
variables in the explicit form.
The following statement follows immediately from the equality (31)
due to its right-hand side is the monogenic function in the domain Π :=
:= {ζ ∈ Ek : fu(ζ) = Du, u = 1, 2, . . . ,m}.
Theorem 2. Every monogenic function Φ: Ω −→ Am
n can be continued
to a monogenic function in the domain Π.
The next statement is a fundamental consequence of the equality (31).
It is true for any domain Ω.
Theorem 3. Let fu(Ek) = C for all u = 1, 2, . . . ,m. Then for every
monogenic function Φ: Ω −→ Am
n in an arbitrary fixed domain Ω, the
Gateaux r-th derivatives Φ(r) are monogenic functions in Ω for all r.
The proof is completely analogous to the proof of Theorem 4 [16].
Using the integral expression (23) of monogenic function Φ : Ω → Am
n
in the case where a domain Ω is convex with respect to the set of directions
Mu , u = 1, 2, . . . ,m, we obtain the following expression for the Gateaux
r-th derivative Φ(r):
Φ(r)(ζ) =
m∑
u=1
Iu
r!
2πi
∫
Γu
Fu(t)
(
(te1 − ζ)−1
)r+1
dt+
264 V. S. Shpakivskyi
+
n∑
s=m+1
Is
r!
2πi
∫
Γus
Gs(t)
(
(te1 − ζ)−1
)r+1
dt ∀ ζ ∈ Ω .
6. Remarks. We note that in the cases where the algebra Am
n has some
specific properties (for instance, properties described in Propositions 1 and 2),
it is easy to simplify the form of the equality (31).
1. Under conditions of Proposition 1 the following equalities hold: um+1 =
um+2 = . . . = un =: η, the representation (31) gets the form
Φ(ζ) =
m∑
u=1
Fu(ξu)Iu +
n∑
s=m+1
s−m+1∑
r=2
1
(r − 1)!
Qr,s F
(r−1)
η (ξη) Is+
+
n∑
s=m+1
Gs(ξη)Is +
n∑
q=m+1
n∑
s=m+1
s−m+1∑
r=2
1
(r − 1)!
Qr,s G
(r−1)
q (ξη) Is Iq . (32)
The formula (32) generalizes representations of monogenic functions in both
three-dimensional harmonic algebras (see [16–18]) and specific n-dimensional
algebras (see [21, 22]) to the case of algebras of more general form and to a
variable of more general form.
2. Under conditions of Proposition 2 the representation (23) gets the form
Φ(ζ) =
m∑
u=1
Fu(ξu)Iu +
n∑
s=m+1
Gs(ξus)Is +
n∑
s=m+1
TsF
′
us
(ξus)Is . (33)
The formula (33) generalizes representations of monogenic functions in both
a three-dimensional harmonic algebra with one-dimensional radical (see [17]) and
semi-simple algebras (see [18, 22]) to the case of algebras of more general form
and to a variable of more general form.
3. Let n = m. Then the algebra An
n is semi-simple and contains no
nilpotent subalgebra. Then the formulae (32), (33) combine to the form Φ(ζ) =
n∑
u=1
Fu(ξu)Iu , because there are no vectors {Ik}nk=m+1. This formula is obtained
in the paper [22].
7. Relations between monogenic functions and
partial differential equations. Consider the following linear
Monogenic functions in finite-dimensional commutative ... 265
partial differential equation with constant coefficients:
LNU(x1, x2, . . . , xk) :=
∑
α1+α2...+αk=N
Cα1,α2,...,αk
∂NΦ
∂xα1
1 ∂xα2
2 . . . ∂xαk
k
= 0,
(34)
If a function Φ(ζ) is N -times differentiable in the sense of Gateaux
in Ω, then ∂α1+α2+...+αkΦ
∂x
α1
1 ∂x
α2
2 ...∂x
αk
k
= eα1
1 eα2
2 . . . eαk
k Φ(α1+α2+...+αk)(ζ) =
= eα2
2 eα3
3 . . . eαk
k Φ(N)(ζ). Therefore, due to the equality
LNΦ(ζ) = Φ(N)(ζ)
∑
α1+α2+...+αk=N
Cα1,α2,...,αk
eα2
2 eα3
3 . . . eαk
k , (35)
every N -times differentiable in the sense of Gateaux in Ω function Φ
satisfies the equation LNΦ(ζ) = 0 in Ω if and only if∑
α1+α2+...+αk=N
Cα1,α2,...,αk
eα2
2 eα3
3 . . . eαk
k = 0. (36)
Really, it follows from (36) that real-valued components
ReUk(x1, x2, . . . , xk) and ImUk(x1, x2, . . . , xk) of the decomposition
(8) are solutions of the equation (34).
In the case where fu(Ek) = C for all u = 1, 2, . . . ,m, it follows from
Theorem 3 that the equality (35) holds for every monogenic function
Φ: Ω −→ Am
n .
Thus, to construct solutions of the equation (34) in the form of
components of monogenic functions, we must to find k linearly independent
over the field R vectors (6) satisfying the characteristic equation (36) and
to verify the condition: fu(Ek) = C for all u = 1, 2, . . . ,m. Then, the
formula (23) gives a constructive description of all mentioned monogenic
functions.
In the next theorem, we assign a special class of equations (34) for
which fu(Ek) = C for all u = 1, 2, . . . ,m. Let us introduce the polynomial
P (b2, b3, . . . , bk) :=
∑
α1+α2+...+αk=N
Cα1,α2,...,αk
bα2
2 bα3
3 . . . bαk
k . (37)
Theorem 4. Suppose that there exist linearly independent over the
field R vectors e1 = 1, e2, . . . , ek in Am
n of the form (6) that satisfy
266 V. S. Shpakivskyi
the equality (36). If P (b2, b3, . . . , bk) ̸= 0 for all real b2, b3, . . . , bk, then
fu(Ek) = C for all u = 1, 2, . . . ,m.
Proof. Using the multiplication table of Am
n we obtain the equalities
eα2
2 =
m∑
u=1
aα2
2u Iu + ΨR , . . . , eαk
k =
m∑
u=1
aαk
ku Iu + ΘR , where
ΨR , . . . ,ΘR ∈ R. Now the equality (36) gets the form
∑
α1+α2+...+αk=N
Cα1,α2,...,αk
(
m∑
u=1
aα2
2u . . . aαk
ku Iu + Ψ̃R
)
= 0, (38)
where Ψ̃R ∈ R. Moreover, due to the assumption that the vectors
e1, e2, . . . , ek of the form (6) satisfy the equality (36), exist complex
coefficients ajr for j = 1, 2, . . . , k, r = 1, 2, . . . , n that satisfy the equality
(38).
It follows from the equality (38) that
∑
α1+α2+...+αk=N
Cα1,α2,...,αk
aα2
2u . . . aαk
ku = 0, u = 1, 2, . . . ,m. (39)
Since P (b2, b3, . . . , bk) ̸= 0 for all {b2, b3, . . . , bk} ⊂ R, the equalities
(39) can be satisfied only if for each u = 1, 2, . . . ,m at least one of
the numbers a2u, a3u, . . . , aku belongs to C \ R that implies the relation
fu(Ek) = C for all u = 1, 2, . . . ,m. The theorem is proved.
We note that if P (b2, b3, . . . , bk) ̸= 0 for all {b2, b3, . . . , bk} ⊂ Rk, then
CN,0,0,...,0 ̸= 0, because otherwise P (b2, b3, . . . , bk) = 0 for b2 = b3 = . . . =
= bk = 0.
Since the function P (b2, b3, . . . , bk) is continuous on Rk, the
condition P (b2, b3, . . . , bk) ̸= 0 means either P (b2, b3, . . . , bk) > 0 or
P (b2, b3, . . . , bk) < 0 for all real b2, b3, . . . , bk. Therefore, it is obvious that
for any equation (34) of the elliptic type, the condition P (b2, b3, . . . , bk) ̸= 0
is always satisfied for all {b2, b3, . . . , bk} ⊂ Rk. At the same time, exist
equations (34) for which P (b2, b3, . . . , bk) > 0 for all {b2, b3, . . . , bk} ⊂ R,
but which are not elliptic. For example, such is the following equation in
R4: ∂3u
∂x3
1
+ ∂3u
∂x1∂x2
2
+ ∂3u
∂x1∂x2
3
+ ∂3u
∂x1∂x2
4
= 0.
Monogenic functions in finite-dimensional commutative ... 267
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|
| id | oai:trim.imath.kiev.ua:article-176 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:03:33Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/a4/192b708b7f8477d8b336d08f1de116a4.pdf |
| spelling | oai:trim.imath.kiev.ua:article-1762018-01-23T12:10:05Z Monogenic functions in finite-dimensional commutative associative algebras Monogenic functions in finite-dimensional commutative associative algebras Shpakivskyi, V. S. Shpakivskyi, V. S. Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra overthe field of complex numbers with $m$ idempotents. Let $e_1=1$,\break $e_2,\ldots,e_k$, $2\leqk\leq 2n$, are linearly independent over the field of real numbers elements of $\mathbb{A}_n^m$.We consider monogenic (i.~e., continuous and differentiable in the sense of Gateaux) functions ofthe variable $\sum_{j=1}^k x_j\,e_j$\,, where $x_1,x_2,\ldots,x_k$ are real, and obtain aconstructive description of all mentioned functions by means of holomorphic functions of complexvariables. Due to this description obtain, that monogenic functions have Gateaux derivatives ofall orders. The present article is a generalization of the author's paper \cite{Shpakivskyi-2014},where mentioned results are obtained for $k=3$. Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra overthe field of complex numbers with $m$ idempotents. Let $e_1=1$,\break $e_2,\ldots,e_k$, $2\leqk\leq 2n$, are linearly independent over the field of real numbers elements of $\mathbb{A}_n^m$.We consider monogenic (i.~e., continuous and differentiable in the sense of Gateaux) functions ofthe variable $\sum_{j=1}^k x_j\,e_j$\,, where $x_1,x_2,\ldots,x_k$ are real, and obtain aconstructive description of all mentioned functions by means of holomorphic functions of complexvariables. Due to this description obtain, that monogenic functions have Gateaux derivatives ofall orders. The present article is a generalization of the author's paper \cite{Shpakivskyi-2014},where mentioned results are obtained for $k=3$. Інститут математики НАН України 2015-04-23 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/176 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 12 No. 3 (2015): Analysis and Applications; 251-268 Сборник Трудов Института математики НАН Украины; Том 12 № 3 (2015): Анализ и приложения; 251-268 Збірник Праць Інституту математики НАН України; Том 12 № 3 (2015): Аналіз та застосування; 251-268 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/176/145 Авторське право (c) 2015 Праці Інституту математики НАН України |
| spellingShingle | Shpakivskyi, V. S. Shpakivskyi, V. S. Monogenic functions in finite-dimensional commutative associative algebras |
| title | Monogenic functions in finite-dimensional commutative associative algebras |
| title_alt | Monogenic functions in finite-dimensional commutative associative algebras |
| title_full | Monogenic functions in finite-dimensional commutative associative algebras |
| title_fullStr | Monogenic functions in finite-dimensional commutative associative algebras |
| title_full_unstemmed | Monogenic functions in finite-dimensional commutative associative algebras |
| title_short | Monogenic functions in finite-dimensional commutative associative algebras |
| title_sort | monogenic functions in finite-dimensional commutative associative algebras |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/176 |
| work_keys_str_mv | AT shpakivskyivs monogenicfunctionsinfinitedimensionalcommutativeassociativealgebras AT shpakivskyivs monogenicfunctionsinfinitedimensionalcommutativeassociativealgebras |