On differentiable and monogenic functions in a harmonic algebra
For locally bounded and differentiable in the sense of G\^ateaux functions $\Phi$ given in a three-dimensional commutativeharmonic algebra with two-dimensional radical, we prove the following statement: if the function $\Phi$ domain is convex "in the radical direction" and the diff...
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| author | Plaksa, S. A. Plaksa, S. A. |
| author_facet | Plaksa, S. A. Plaksa, S. A. |
| author_institution_txt_mv | [
{
"author": "S. A. Plaksa",
"institution": "Institute of Mathematics of the National Academy of Sciences of Ukraine"
}
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| description | For locally bounded and differentiable in the sense of G\^ateaux functions $\Phi$ given in a three-dimensional commutativeharmonic algebra with two-dimensional radical, we prove the following statement: if the function $\Phi$ domain is convex "in the radical direction" and the difference $\zeta_1-\zeta_2$ belongs to the radical, the difference $\Phi(\zeta_1)-\Phi(\zeta_2)$ belongs also to the radical. As a result, we prove that locally bounded and differentiable in the sense of G\^ateaux functions are also differentiable in the sense of Lorch. |
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Збiрник праць Iн-ту математики НАН України 2017, том 14, № 1, 210–221
УДК 517.5
S.A. Plaksa
(Institute of Mathematics of the National Academy of Sciences of
Ukraine, Kyiv)
plaksa@imath.kiev.ua
On differentiable and monogenic
functions in a harmonic algebra
Dedicated to Prof. Yu. B. Zelinskii on the occasion of his 70th birthday
For locally bounded and differentiable in the sense of Gâteaux functions
Φ given in a three-dimensional commutative harmonic algebra with two-
dimensional radical, we prove the following statement: if the function Φ
domain is convex "in the radical direction" and the difference ζ1−ζ2 belongs
to the radical, the difference Φ(ζ1) − Φ(ζ2) belongs also to the radical. As
a result, we prove that locally bounded and differentiable in the sense of
Gâteaux functions are also differentiable in the sense of Lorch.
Для локально обмежених i диференцiйовних за Гато функцiй Φ, визна-
чених у тривимiрнiй комутативнiй гармонiчнiй алгебрi з двовимiрним
радикалом, ми доводимо наступне твердження: якщо область визначе-
ння функцiї Φ опукла "у напрямку радикала" i рiзниця ζ1 − ζ2 нале-
жить радикалу, то рiзниця Φ(ζ1)−Φ(ζ2) також належить радикалу. Як
наслiдок, доводиться, що локально обмеженi i диференцiйовнi за Гато
функцiї є також диференцiйовними за Лорхом.
1. Introduction. In the algebra of complex numbers C a function
F : C −→ C is called monogenic at a point ξ0 ∈ C if there exists the finite
limit
lim
ξ→ξ0
F (ξ)− F (ξ0)
ξ − ξ0
(1)
and this limit, which is called the derivative of the function at the point ξ0,
is the same when ξ tends to ξ0 by any way. A function, which is monоgenic
c© S.A. Plaksa, 2017
On differentiable and monogenic functions ... 211
at all points of a domain D ⊂ C, is called holomorphic in this domain.
Class of holomorphic in D functions coincides with a class of analytic in D
functions which are represented in a certain neighborhood of every point
ξ0 ∈ D in the form of the sum of convergent power series (cf. [1]).
Every analytic function F (ξ) of the complex variable ξ = x+iy satisfies
the two-dimensional Laplace equation(
∂2
∂x2 +
∂2
∂y2
)
F (ξ) ≡ F ′′(ξ) (12 + i2) = 0
due to the equality 12 + i2 = 0 for the unit 1 and the imaginary unit i of
the algebra of complex numbers.
An effectiveness of the analytic function methods in the complex plane
for researching plane potential fields inspires mathematicians to develop
analogous methods for spatial fields.
In the paper [2], analytic functions with values in a commutative
algebra different from the algebra C are used for a construction of solutions
of three-dimensional Laplace equation.
Let A be a commutative Banach algebra of a rank n, 3 ≤ n ≤ ∞,
over either the field of real numbers R or the field of complex numbers C.
Let {e1, e2, e3} be a part of the basis of A. P.W. Ketchum [2] has shown
that if linearly independent elements e1, e2, e3 ∈ A satisfy the condition
e2
1 + e2
2 + e2
3 = 0 , (2)
then every analytic function Φ(ζ) of the variable ζ = xe1 + ye2 + ze3 with
real x, y, z satisfies the three-dimensional Laplace equation(
∂2
∂x2 +
∂2
∂y2 +
∂2
∂z2
)
Φ(ζ) ≡ Φ′′(ζ) (e2
1 + e2
2 + e2
3) = 0 , (3)
where Φ′′(ζ) can be understood in a certain sense. An algebra A is called
harmonic (cf. [2–4]) if in A there exists a triad of linearly independent
vectors satisfying the equality (2).
It is clear that a characterization of functions satisfying the equalities
(3) has relation to a question: in what sense the derivative is understood
in the algebra A.
It is well-known that there exist various definitions of differentiable
functions given in algebras. Choosing concepts of a differentiable function
and its derivative, it is natural to desire to combine the largest set of
212 S.A. Plaksa
functions satisfying the equalities (3) with the preservation of the basic
properties of analytic functions of a complex variable for functions of the
mentioned set.
Some properties similar to properties of analytic functions of complex
variable are established for functions differentiable in the sense of Lorch [5]
in an arbitrary convex domain of commutative Banach algebra. In parti-
cular, the integral Cauchy theorem and the integral Cauchy formula, the
Taylor expansion and the Morera theorem are proved in [5] in such a way
as for analytic functions of complex variable. The convexity of domain in
the mentioned results from [5] is withdrawn by E.K. Blum [6].
I. P. Mel’nichenko [7] suggested to consider doubly differentiable in
the sense of Gâteaux functions in the equalities (3). Let us note that a
priori the differentiability of the function Φ in the sense of Gâteaux is a
restriction being weaker than the differentiability of this function in the
sense of Lorch.
To prove analogues of principal theorems of the analytic function theory
in the complex plane, in the papers [8–11] we considered monogenic functi-
ons (i.e., continuous differentiable in the sense of Gâteaux functions) in
some harmonic algebras. We developed the following research scheme: at
first, it is useful to obtain a constructive description of monogenic functi-
ons by means of analytic functions of complex variables; hereupon, to show
that monogenic functions have the continuous Gâteaux derivatives of all
orders and are differentiable in the sense of Lorch as well; and then to prove
integral theorems and to obtain the Taylor and Laurent expansions. In the
papers [12–14] such a scheme is extended to the case of monogenic functi-
ons in an arbitrary finite-dimensional commutative associative algebra.
The initial point of the mentioned research scheme is the following
statement: for a monogenic function Φ, the difference Φ(ζ1)−Φ(ζ2) belongs
to a maximal ideal of a commutative finite-dimensional algebra A if the
function Φ domain is convex "in the direction" of this ideal and the di-
fference ζ1 − ζ2 belongs to the same ideal. For the first time, such a
statement was proved in the papers [8, 9] for monogenic functions in a
three-dimensional harmonic algebra A3 with two-dimensional radical.
In this paper, we prove similar statement for given in A3 functions Φ
which are differentiable in the sense of Gâteaux and locally bounded, i.e.
the assumption from [8, 9] on continuity of Φ is weakened. Obviously, the
proved statement opens a way to similar generalizations of other results
from [8–14].
On differentiable and monogenic functions ... 213
2. A harmonic algebra A3. Let A3 be a three-dimensional
commutative associative Banach algebra with the unit 1 over the field
of complex numbers C. Let {1, ρ1, ρ2} be a basis of the algebra A3 with
the multiplication table
ρ1ρ2 = ρ2
2 = 0, ρ2
1 = ρ2.
The algebra A3 is harmonic. A basis {e1, e2, e3} satisfying the equality
(2) is called harmonic. All harmonic bases in A3 are described in Theorem
1.6 [4]. In particular, the basis {e1, e2, e3} is harmonic if decompositions
of its elements with respect to the basis {1, ρ1, ρ2} are of the form
e1 = 1,
e2 = n1 + n2ρ1 + n3ρ2,
e3 = m1 +m2ρ1 +m3ρ2,
(4)
where nk and mk for k = 1, 2, 3 are complex numbers satisfying the system
of equations
1 + n2
1 +m2
1 = 0,
n1n2 +m1m2 = 0,
xn2
2 +m2
2 + 2(n1n3 +m1m3) = 0
(5)
and the inequality n2m3−n3m2 6= 0, and moreover, at least one of numbers
in each of the pairs (n1, n2) and (m1,m2) is not equal to zero.
The algebra A3 have the unique maximal ideal I := {λ1ρ1 + λ2ρ2 :
λ1, λ2 ∈ C} which is also the radical of A3.
Consider the linear functional f : A3 → C such that the maximal ideal
I is its kernel and f(1) = 1. It is well known [15, p. 135] that f is also a
multiplicative functional, i.e. the equality f(ab) = f(a)f(b) is fulfilled for
all a, b ∈ A3.
We use the euclidian norm ‖a‖ :=
√
|a1|2 + |a2|2 + |a2|2 in the algebra
A3, where a = a1e1 + a2e2 + a3e3 and a1, a2, a3 ∈ C.
3. Differentiability in the sense of Lorch and in the
sense of Gâteaux. Monogenic functions. In what follows,
{e1, e2, e3} is a harmonic basis of the form (4), E3 := {ζ := xe1+ye2+ze3 :
x, y, z ∈ R} is the linear span generated by the vectors e1, e2, e3 and
ζ = xe1 + ye2 + ze3, where x, y, z ∈ R.
Let Ω be a domain in R3. Associate with Ω the congruent domain
Ωζ := {ζ = xe1 + ye2 + ze3 : (x, y, z) ∈ Ω} in E3. Associate similarly with
any set Q ⊂ R3 the set Qζ ⊂ E3.
214 S.A. Plaksa
Consider a function Φ: Ωζ −→ A3 and properties of differentiability of
such a function.
The concepts of Fréchet derivative and Gâteaux derivative are used
for mappings of linear normalized spaces. These derivatives are defined as
linear operators. In the considered case, they are linear operators from E3
into A3.
For a mapping given in a domain of a commutative Banach algebra,
E.R. Lorch [5] introduced a derivative, which is understood as a function
given in the same domain.
A function Φ: Ωζ −→ A3 is called differentiable in the sense of Lorch
(cf. [5]) in a domain Ωζ ⊂ E3 if for every ζ ∈ Ωζ there exists an element
Φ′L(ζ) ∈ A3 such that for any ε > 0 there exists δ > 0 such that for all
h ∈ E3 with ‖h‖ < δ the following inequality fulfilled:
‖Φ(ζ + h)− Φ(ζ)− hΦ′L(ζ)‖ ≤ ‖h‖ ε . (6)
Obviously, in the inequality (6) the Lorch derivative Φ′L(ζ) is a function of
the variable ζ, i.e., Φ′L : Ωζ −→ A3 .
At the same time, the mapping Bζ : E3 −→ A3, which is defined by
the equality Bζh := hΦ′L(ζ), is a bounded linear operator. Therefore, a
function Φ, which is differentiable in the sense of Lorch in a domain Ωζ ,
have the Fréchet derivative Bζ in every point ζ ∈ Ωζ (cf. [15, p. 115]). The
converse is not true, see an example in [15, p. 116].
Using the Gâteaux differential, I. P. Mel’nichenko [7] suggested to consi-
der the Gâteaux derivative as a function Φ′G : Ωζ −→ A3 too.
We say that a function Φ: Ωζ −→ A3 is called differentiable in the
sense of Gâteaux in a domain Ωζ ⊂ E3 if for every ζ ∈ Ωζ there exists an
element Φ′G(ζ) ∈ A3 such that
lim
δ→0+0
(Φ(ζ + δh)− Φ(ζ)) δ−1 = hΦ′G(ζ) ∀h ∈ E3. (7)
Obviously, the Gâteaux derivative Φ′G(ζ) is a function of the variable ζ and
is a generalization of the classical directional derivative.
The left-hand side of the equality (7) is called the Gâteaux differential
of function Φ. It is well-known, in a general case, the Gâteaux differential
may fail to be linear with respect to h. But, it is clear, if the Gâteaux
derivative Φ′G(ζ) exists, the Gâteaux differential (7) is a bounded linear
operator with respect to h. At the same time, the converse is not true as
the same example in [15, p. 116] shows.
On differentiable and monogenic functions ... 215
It is evident, the definition (6) of the Lorch derivative and the definition
(7) of the Gâteaux derivative take into account the existence of noninverti-
ble elements h in the algebra A3 because the division by elements of algebra
is not used in them in contrast to the classical definition (1) of complex
derivative.
Obviously, if a function Φ is differentiable in the sense of Lorch in Ωζ ,
then it is also differentiable in the sense of Gâteaux, and Φ′L(ζ) = Φ′G(ζ)
for all ζ ∈ Ωζ . The converse is clearly not true similarly to the fact that
the existence of all directional derivatives at a point does not guarantee a
strong differentiability (or even continuity) of function at that point.
Let us consider a concept of monogenic function Φ: Ωζ −→ A3.
We say that a function Φ: Ωζ −→ A3 ismonogenic in a domain Ωζ ⊂ E3
if Φ is continuous and differentiable in the sense of Gâteaux at every point
of Ωζ .
We use the notion of monogenic function in the sense of existence of
derived numbers for this function (cf. [1,16]). In the scientific literature the
denomination of monogenic function is used else for functions satisfying
certain conditions similar to the classical Cauchy – Riemann conditions
(cf. [17, 18]). Such functions are also called regular functions (cf. [19]) or
hyperholomorphic functions (cf. [20, 21]).
In the paper [8] we obtained a constructive description of monogenic
functions by means of analytic functions of complex variables (see also [9]).
As a consequence of such a description, every monogenic in Ωζ function is
differentiable in the sense of Lorch in Ωζ .
To explain it, without loss of generality, we assume that a function
Φ: Ωζ −→ A3 is monogenic in a convex domain Ωζ (if Ωζ is not convex, it
is possible to consider a restriction of the function Φ to any ball lying in
Ωζ). Denote D := {ξ = f(ζ) : ζ ∈ Ωζ}. Then for every monogenic function
Φ: Ωζ −→ A3 there exist complex-valued analytic functions F , F1, F2 in
the domain D such that (see [8, 9])
Φ(ζ) = F (ξ)+
(
F1(ξ)+(n2y+m2z)F
′(ξ)
)
ρ1+
(
F2(ξ)+(n2y+m2z)F
′
1(ξ)+
+ (n3y +m3z)F
′(ξ) +
(n2y +m2z)
2
2
F ′′(ξ)
)
ρ2 ∀ ζ ∈ Ωζ , (8)
where ξ = x+n1y+m1z. The equality (8) can be rewritten in the following
216 S.A. Plaksa
form (see [9])
Φ(ζ) =
1
2πi
∫
Γζ
(
F (t) + ρ1F1(t) + ρ2F2(t)
)
(t− ζ)−1dt ∀ζ ∈ Ωζ , (9)
where Γζ is an arbitrary closed Jordan rectifiable curve in D, which is
homotopic to the point f(ζ) and embraces this point.
It follows from the equality (9) that the function Φ is differentiable in
the sense of Lorch in Ωζ . Using the equality (9), we obtain the following
expression for the Lorch n-th derivative, which coincides with the Gateaux
n-th derivative:
Φ(n)(ζ) =
n!
2πi
∫
Γζ
(
F (t)+ρ1F1(t)+ρ2F2(t)
)(
(t−ζ)−1
)n+1
dt ∀ζ ∈ Ωζ .
Thus, every monogenic function Φ satisfies the equalities (3).
Bellow, we show that similar statements are true for functions Φ which
are differentiable in the sense of Gâteaux and locally bounded in Ωζ , i.e.,
the assumption from [8,9] on continuity of Φ will be weakened.
4. Some special properties of locally bounded and
differentiable in the sense of Gâteaux functions. Consider
a function Φ: Ωζ −→ A3 which is differentiable in the sense of Gâteaux
in a domain Ωζ . It is follows from Theorem 1.3 in [4] that the function Φ
satisfies the following conditions in Ωζ :
∂Φ
∂y
=
∂Φ
∂x
e2,
∂Φ
∂z
=
∂Φ
∂x
e3. (10)
All noninvertible elements in A3 belong to the radical I, which is the
kernel of functional f . Therefore, an element ζ = x + ye2 + ze3 ∈ E3 is
noninvertible in A3 if and only if the point (x, y, z) belongs to the following
straight line in R3 (see [8, 9]):
L :
{
x+ yRen1 + zRem1 = 0,
yImn1 + zImm1 = 0 .
We say that a domain Ω ⊂ R3 is convex in the direction of the straight
line L if Ω contains every segment parallel to L and connecting two points
(x1, y1, z1), (x2, y2, z2) ∈ Ω. It may be said that the congruent domain Ωζ
is convex "in the radical direction".
On differentiable and monogenic functions ... 217
Let us prove the following statement for a function Φ: Ωζ −→ A3 which
is differentiable in the sense of Gâteaux and locally bounded in a domain
Ωζ which is convex "in the radical direction".
Lemma 1. Let a domain Ω ⊂ R3 be convex in the direction of the straight
line L and Φ: Ωζ −→ A3 be a locally bounded and differentiable in the sense
of Gâteaux function in the domain Ωζ . If ζ1, ζ2 ∈ Ωζ and ζ2 − ζ1 ∈ Lζ ,
then
Φ(ζ1)− Φ(ζ2) ∈ I. (11)
Proof. Inasmuch as f is a linear continuous multiplicative functional,
from the equalities (10) it follows that
f
(
∂Φ
∂y
)
= f
(
∂Φ
∂x
)
f (e2) , f
(
∂Φ
∂z
)
= f
(
∂Φ
∂x
)
f (e3) . (12)
Consider the decomposition
Φ(ζ) = V0(x, y, z) + V1(x, y, z) ρ1 + V2(x, y, z) ρ2 , (13)
of function Φ: Ωζ −→ A3 with respect to the basis {1, ρ1, ρ2}.
Substituting the expressions (4), (13) into the equalities (12), we get
the following equalities:
∂V0
∂y
= n1
∂V0
∂x
,
∂V0
∂z
= m1
∂V0
∂x
. (14)
Inasmuch as
ξ = f(ζ) = (x+ yRen1 + zRem1) + i(yImn1 + zImm1) =: τ + iη , (15)
from the equalities (14) we get
∂V0
∂η
Imn1 = i
∂V0
∂τ
Imn1 ,
∂V0
∂η
Imm1 = i
∂V0
∂τ
Imm1 . (16)
It follows from the first equation of the system (5) that, at least one
of the numbers Imn1, Imm1 is not equal to zero. Therefore, from (16) we
get the equality
∂V0(x, y, z)
∂η
= i
∂V0(x, y, z)
∂τ
(17)
218 S.A. Plaksa
which is fulfilled for all (x, y, z) ∈ Ω .
Let us prove that V0(x1, y1, z1) = V0(x2, y2, z2) for the points
(x1, y1, z1), (x2, y2, z2) ∈ Ω such that the segment connecting these points
is parallel to the straight line L.
Let us construct in Ω two surfaces Q and Σ satisfying the following
conditions:
• Q and Σ have the same edge;
• the surface Q contains the point (x1, y1, z1) and the surface Σ contai-
ns the point (x2, y2, z2);
• restrictions of the functional f onto the sets Qζ and Σζ are one-to-
one mappings of these sets onto the same domain G of the complex
plane.
As the surface Q, we can take an equilateral triangle having the center
(x1, y1, z1) and apexes A1, A2, A3, and, in addition, the plane of this tri-
angle is perpendicular to the straight line L.
To construct the surface Σ, first, consider a triangle with the center
(x2, y2, z2) and apexes A′1, A′2, A′3 such that the segments A′1A′2, A′2A′3,
A′1A
′
3 are parallel to the segments A1A2, A2A3, A1A3, respectively, and,
in addition, the length of A′1A′2 is less than the length of A1A2. Inasmuch
as the domain Ω is convex in the direction of the straight line L, the prism
with vertexes A′1, A′2, A′3, A′′1 , A′′2 , A′′3 is completely contained in Ω, where
the points A′′1 , A′′2 , A′′3 are located in the plane of triangle A1A2A3 and the
edges A′mA′′m are parallel to L for m = 1, 3.
Further, set a triangle with apexes B1, B2, B3 such that the point Bm
is located on the segment A′mA′′m for m = 1, 3 and the truncated pyramid
with vertexes A1, A2, A3, B1, B2, B3 and lateral edges AmBm, m = 1, 3, is
completely contained in the domain Ω.
At last, in the plane of triangle A′1A′2A′3 set a triangle T with apexes
C1, C2, C3 such that the segments C1C2, C2C3, C1C3 are parallel to the
segments A′1A′2, A′2A′3, A′1A′3, respectively, and, in addition, the length of
C1C2 is less than the length of A′1A′2. It is evident that the truncated
pyramid with vertexes B1, B2, B3, C1, C2, C3 and lateral edges BmCm,
m = 1, 3, is completely contained in the domain Ω.
Now, as the surface Σ, denote the surface formed by the triangle T and
the lateral surfaces of mentioned truncated pyramids A1A2A3B1B2B3 and
B1B2B3C1C2C3.
On differentiable and monogenic functions ... 219
For each ξ ∈ G define two complex valued functions H1 and H2 so that
H1(ξ) := f(Φ(ζ)) ≡ V0(x, y, z) for (x, y, z) ∈ Q,
H2(ξ) := f(Φ(ζ)) ≡ V0(x, y, z) for (x, y, z) ∈ Σ,
where the correspondence between the points (x, y, z) and ξ ∈ G is determi-
ned by the relation (15). The functions H1, H2 are analytic in the domain
G due to the equality (17) and Theorem 6 [22].
Inasmuch as H1, H2 are continuous in the closure of domain G and
H1(ξ) ≡ H2(ξ) on the boundary of G, this identity is fulfilled everywhere
in G. Therefore, V0(x1, y1, z1) = V0(x2, y2, z2) and the equalities
f(Φ(ζ2)− Φ(ζ1)) = f(Φ(ζ2))− f(Φ(ζ1)) = 0,
are fulfilled for ζ1 := x1e1 + y1e2 + z1e3 and ζ2 := x2e1 + y2e2 + z2e3.
Thus, Φ(ζ2)−Φ(ζ1) belongs to the kernel I of functional f . The lemma is
proved.
Now, using Lemma, in such a way as in the papers [8,9], we obtain the
expression (8) or, that is the same, the expression (9) for a locally bounded
and differentiable in the sense of Gâteaux function Φ : Ωζ → A3 in the
case where a domain Ω is convex in the direction of the straight line L.
As a result, we obtain the following statement.
Theorem 1. For a function Φ: Ωζ −→ A3 given in an arbitrary domain
Ωζ ⊂ E3 the following properties are equivalent:
(I) Φ is a locally bounded and differentiable in the sense of Gâteaux
function in Ωζ ;
(II) Φ is a monogenic function in Ωζ ;
(III) Φ is a differentiable in the sense of Lorch function in Ωζ .
Certainly, the property of function to be locally bounded and differenti-
able in the sense of Gâteaux in Ωζ is also equivalent to the various defini-
tions of monogenic function, that are stated in Theorem 1.15 [9].
This research is partially supported by the Ministry of Education and
Science of Ukraine (Project No. 0116U001528).
220 S.A. Plaksa
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
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| language | English |
| last_indexed | 2026-08-04T01:00:53Z |
| publishDate | 2017 |
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| spelling | oai:trim.imath.kiev.ua:article-1142018-02-13T13:58:16Z On differentiable and monogenic functions in a harmonic algebra Plaksa, S. A. Plaksa, S. A. For locally bounded and differentiable in the sense of G\^ateaux functions $\Phi$ given in a three-dimensional commutativeharmonic algebra with two-dimensional radical, we prove the following statement: if the function $\Phi$ domain is convex "in the radical direction" and the difference $\zeta_1-\zeta_2$ belongs to the radical, the difference $\Phi(\zeta_1)-\Phi(\zeta_2)$ belongs also to the radical. As a result, we prove that locally bounded and differentiable in the sense of G\^ateaux functions are also differentiable in the sense of Lorch. Для локально обмежених і диференційовних за Гато функцій $\Phi$, визначених в тривимірній комутативній гармонічній алгебрі здвовимірним радикалом ми доводимо наступне твердження: якщо область визначення функції $\Phi$ опукла "у напрямку радикала" і різниця $\zeta_1-\zeta_2$ належить радикалу, то різниця $\Phi(\zeta_1)-\Phi(\zeta_2)$ також належить радикалу. Як наслідок, доводиться, що локально обмежені і диференційовні за Гато функції є також диференційовними за Лорхом. Інститут математики НАН України 2017-04-25 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/114 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 14 No. 1 (2017): Vol. 14 No. 1 (2017): Analysis and Applications; 210–221 Сборник Трудов Института математики НАН Украины; Том 14 № 1 (2017): Том 14 № 1 (2017): Анализ и приложения; 210–221 Збірник Праць Інституту математики НАН України; Том 14 № 1 (2017): Аналіз та застосування; 210–221 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/114/103 Авторське право (c) 2017 Праці Інституту математики НАН України |
| spellingShingle | Plaksa, S. A. Plaksa, S. A. On differentiable and monogenic functions in a harmonic algebra |
| title | On differentiable and monogenic functions in a harmonic algebra |
| title_full | On differentiable and monogenic functions in a harmonic algebra |
| title_fullStr | On differentiable and monogenic functions in a harmonic algebra |
| title_full_unstemmed | On differentiable and monogenic functions in a harmonic algebra |
| title_short | On differentiable and monogenic functions in a harmonic algebra |
| title_sort | on differentiable and monogenic functions in a harmonic algebra |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/114 |
| work_keys_str_mv | AT plaksasa ondifferentiableandmonogenicfunctionsinaharmonicalgebra AT plaksasa ondifferentiableandmonogenicfunctionsinaharmonicalgebra |