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Commutative Algebras Associated with Classic Equations of Mathematical Physics

August 2012

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Institute of Mathematics

The idea of an algebraic-analytic approach to equations of mathematical physics means to find a commutative Banach algebra such that monogenic functions with values in this algebra have components satisfying to given equations with partial derivatives. We obtain here a constructive description of monogenic functions taking values in a commutative algebra associated with a two-dimensional biharmonic equation by means of analytic functions of complex variables. For the mentioned monogenic functions we establish basic properties analogous to properties of analytic functions of complex variables: the Cauchy integral theorem and integral formula, the Morera theorem, the uniqueness theorem, and the Taylor and Laurent expansions. Similar results are obtained for monogenic functions which take values in a three-dimensional commutative algebra and satisfy the three-dimensional Laplace equation. In infinite-dimensional commutative Banach algebras we construct explicitly monogenic functions which have components satisfying the threedimensional Laplace equation. We establish that all spherical functions are components of the mentioned monogenic functions. A relation between these monogenic functions and harmonic vectors is described. We establish that solutions of elliptic equations degenerating on an axis are constructed by means of components of analytic functions taking values in an infinite-dimensional commutative Banach algebra. In such a way we obtain integral expressions for axial-symmetric potentials and Stokes flow functions in an arbitrary simply connected domain symmetric with respect to an axis.

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Advances in Applied Analysis

Trends in Mathematics, 177–223

2012 Springer Basel AG

Commutative Algebras Associated with

Classic Equations of Mathematical Physics

S.A. Plaksa

Abstract. The idea of an algebraic-analytic approach to equations of mathe-

matical physics means to ﬁnd a commutative Banach algebra such that mono-

genic functions with values in this algebra have components satisfying to given

equations with partial derivatives.

We obtain here a constructive description of monogenic functions tak-

ing values in a commutative algebra associated with a two-dimensional bi-

harmonic equation by means of analytic functions of complex variables. For

the mentioned monogenic functions we establish basic properties analogous

to properties of analytic functions of complex variables: the Cauchy integral

theorem and integral formula, the Morera theorem, the uniqueness theorem,

and the Taylor and Laurent expansions. Similar results are obtained for mono-

genic functions which take values in a three-dimensional commutative algebra

and satisfy the three-dimensional Laplace equation.

In inﬁnite-dimensional commutative Banach algebras we construct ex-

plicitly monogenic functions which have components satisfying the three-

dimensional Laplace equation. We establish that all spherical functions are

components of the mentioned monogenic functions. A relation between these

monogenic functions and harmonic vectors is described.

We establish that solutions of elliptic equations degenerating on an axis

are constructed by means of components of analytic functions taking values in

an inﬁnite-dimensional commutative Banach algebra. In such a way we obtain

integral expressions for axial-symmetric potentials and Stokes ﬂow functions

in an arbitrary simply connected domain symmetric with respect to an axis.

Mathematics Subject Classiﬁcation (2010). Primary 30G35; Secondary 35J05,

31A30.

Keywords. Laplace equation; biharmonic equation; harmonic vector; axial-

symmetric potential; Stokes ﬂow function; harmonic commutative Banach al-

gebra; biharmonic algebra; monogenic function; Cauchy–Riemann conditions;

Cauchy integral theorem; Cauchy integral formula; Morera theorem; Taylor

expansion; Laurent expansion.

178 S.A. Plaksa

1. Algebras associated with the Laplace equation

1.1. Spatial stationary potential solenoid vector ﬁeld

Consider a spatial stationary vector ﬁeld deﬁned by means of the vector-function

V≡V(𝑥, 𝑦, 𝑧 ) of the Cartesian coordinates 𝑥, 𝑦, 𝑧 . The vector Vis deﬁned by

means of three real scalar functions 𝑣1:= 𝑣1(𝑥, 𝑦, 𝑧), 𝑣2:= 𝑣2(𝑥, 𝑦, 𝑧 ), 𝑣3:=

𝑣3(𝑥, 𝑦, 𝑧) which give its coordinates in the point (𝑥, 𝑦, 𝑧), videlicet: V=(𝑣1,𝑣

2,𝑣

3).

Deﬁning a potential solenoid ﬁeld in a simply connected domain 𝑄of the

three-dimensional real space ℝ3, the vector-function Vsatisﬁes the system of equa-

tions

div V=0,rot V=0,(1.1)

that we rewrite also in expanded form:

∂𝑣1

∂𝑥 +∂𝑣2

∂𝑦 +∂𝑣3

∂𝑧 =0,

∂𝑣3

∂𝑦 −∂𝑣2

∂𝑧 =0,

∂𝑣1

∂𝑧 −∂𝑣3

∂𝑥 =0,

∂𝑣2

∂𝑥 −∂𝑣1

∂𝑦 =0.

(1.2)

Then there exists a scalar potential function 𝑢(𝑥, 𝑦, 𝑧) such that

V=grad𝑢:= ∂𝑢

∂𝑥 ,∂𝑢

∂𝑦 ,∂𝑢

∂𝑧 ,

and 𝑢satisﬁes the three-dimensional Laplace equation

Δ3𝑢(𝑥, 𝑦, 𝑧):=∂2

∂𝑥2+∂2

∂𝑦2+∂2

∂𝑧2𝑢(𝑥, 𝑦, 𝑧)=0.(1.3)

Doubly continuously diﬀerentiable functions satisfying equation (1.3) are

called harmonic functions, and solutions of the system (1.1) are called harmonic

vectors.

Every harmonic function 𝑢(𝑥, 𝑦, 𝑧) generates a harmonic vector V=grad𝑢,

and the coordinates of vector V=(𝑣1,𝑣

2,𝑣

3) are solutions of (1.3).

1.2. A relation between the two-dimensional Laplace equation

and the algebra of complex numbers

An important achievement of mathematics is the description of plane potential

ﬁelds by means of analytic functions of a complex variable.

A potential 𝑢(𝑥, 𝑦) and a ﬂow function 𝑣(𝑥, 𝑦) of plane stationary potential

solenoid ﬁeld satisfy the Cauchy–Riemann conditions

∂𝑢(𝑥, 𝑦)

∂𝑥 =∂𝑣(𝑥, 𝑦)

∂𝑦 ,∂𝑢(𝑥, 𝑦)

∂𝑦 =−∂𝑣(𝑥, 𝑦)

∂𝑥 ,

Commutative Algebras and Equations of Mathematical Physics 179

and they form the complex potential 𝐹(𝑥+𝑖𝑦)=𝑢(𝑥, 𝑦)+𝑖𝑣(𝑥, 𝑦 ), being an analytic

function of complex variable 𝑥+𝑖𝑦. In turn, every analytic function 𝐹(𝑥+𝑖𝑦)

satisﬁes the two-dimensional Laplace equation

Δ2𝐹:= ∂2𝐹

∂𝑥2+∂2𝐹

∂𝑦2≡𝐹′′(𝑥+𝑖𝑦)(1

2+𝑖2)=0

owing to the equality 12+𝑖2= 0 for the unit 1 and the imaginary unit 𝑖of the

algebra of complex numbers.

1.3. Attempts to ﬁnd an algebra associated with the three-dimensional Laplace

equation

Eﬀectiveness of analytic function methods in the complex plane for researching

plane potential ﬁelds has inspired mathematicians to develop analogous methods

for spatial ﬁelds.

Apparently, W. Hamilton (1843) made the ﬁrst attempts to construct an

algebra associated with the three-dimensional Laplace equation (1.3) in the sense

that components of hypercomplex functions satisfy (1.3). However, Hamilton’s

quaternions form a noncommutative algebra, and after constructing the quaternion

algebra he made no attempt to construct any other algebra (see [1]).

1.4. Harmonic triads in commutative algebras. Harmonic algebras

Let 𝔸be a commutative associative Banach algebra of a rank 𝑛(3 ≤𝑛≤∞)over

either the ﬁeld of real numbers ℝor the ﬁeld of complex numbers ℂ.Let{𝑒1,𝑒

2,𝑒

be a part of the basis of 𝔸and 𝐸3:= {𝜁:= 𝑥𝑒1+𝑦𝑒2+𝑧𝑒3:𝑥, 𝑦, 𝑧 ∈ℝ}be the

linear envelope generated by the vectors 𝑒1,𝑒

2,𝑒

A function Φ : 𝑄𝜁→𝔸is analytic in a domain 𝑄𝜁⊂𝐸3if in a certain

neighborhood of every point 𝜁0∈𝑄𝜁it can be represented in the form of the sum

of convergent power series with coeﬃcients belonging to the algebra 𝔸:

Φ(𝜁)= ∞



𝑘=0

𝑐𝑘(𝜁−𝜁0)𝑘,𝑐

𝑘∈𝔸.(1.4)

It is obvious that if the basic elements 𝑒1,𝑒

2,𝑒

3satisfy the condition

𝑒2

1+𝑒2

2+𝑒2

3=0,(1.5)

then every analytic function Φ : 𝑄𝜁→𝔸satisﬁes equation (1.3), because

Δ3Φ(𝜁)≡Φ′′(𝜁)(𝑒2

1+𝑒2

2+𝑒2

3)=0,𝜁=𝑥𝑒1+𝑦𝑒2+𝑧𝑒3,(1.6)

where Φ′′(𝜁) can be obtained by a formal double diﬀerentiation of the series (1.4),

i.e., Φ′′(𝜁)= ∞



𝑘=0

𝑘(𝑘−1) 𝑐𝑘(𝜁−𝜁0)𝑘−2.

We say tha t an alg e b r a 𝔸is harmonic (see [2–4]) if in 𝔸there exists a triad

of linearly independent vectors {𝑒1,𝑒

2,𝑒

3}satisfying the equality (1.5) provided

that 𝑒2

𝑘∕=0for𝑘=1,2,3. We say also that such a triad {𝑒1,𝑒

2,𝑒

3}is harmonic.

180 S.A. Plaksa

P.W. Ketchum [2] considered the C. Segre algebra of quaternions [5] in its

relations with the three-dimensional Laplace equation. Indeed, in the Segre alge-

bra of quaternions there is the unit 1, and the multiplication table for the basis

{1,𝑖,𝑗,𝑘}is of the following form:

𝑖2=𝑗2=−1,𝑘

2=1,𝑖𝑗=𝑘, 𝑖 𝑘 =−𝑗, 𝑗 𝑘 =−𝑖.

Therefore, there are harmonic triads, in particular: 𝑒1=√2,𝑒

2=𝑖, 𝑒3=𝑗.

K.S. Kunz [6] developed a method for a formal construction of solutions (1.3)

by using power series in any harmonic algebra over the ﬁeld ℂ.

1.5. Diﬀerentiability in the sense of Gateaux. Monogenic functions

I. P. Mel’nichenko [7] noticed that doubly diﬀerentiable, in the sense of Gateaux,

functions form the largest class of functions Φ satisfying identically the equality

(1.6), where Φ′′ is the Gateaux second derivative of the function Φ.

We say that a continuous function Φ : Ω𝜁→𝔸is monogenic in a domain

Ω𝜁⊂𝐸3if Φ is diﬀerentiable in the sense of Gateaux in every point of Ω𝜁, i.e., if

for every 𝜁∈Ω𝜁there exists an element Φ′(𝜁)∈𝔸such that

lim

𝜀→0+0 (Φ(𝜁+𝜀ℎ)−Φ(𝜁)) 𝜀−1=ℎΦ′(𝜁)∀ℎ∈𝐸3.(1.7)

Thus, if the basic elements 𝑒1,𝑒

2,𝑒

3satisfy the condition (1.5), then every

doubly diﬀerentiable, in the sense of Gateaux, function Φ : Ω𝜁→𝔸satisﬁes the

equality (1.6) in the domain Ω𝜁. In turn, if there exists a doubly diﬀerentiable, in

the sense of Gateaux, function Φ : Ω𝜁→𝔸satisfying the equality (1.6) and the

inequality Φ′′(𝜁)∕= 0 at least at one point 𝜁:= 𝑥𝑒1+𝑦𝑒2+𝑧𝑒3∈Ω𝜁, then in this

case the condition (1.5) is satisﬁed.

I.P. Mel’nichenko suggested an algebraic-analytic approach to equations of

mathematical physics, which means to ﬁnd a commutative Banach algebra such

that diﬀerentiable, in the sense of Gateaux, functions with values in this algebra

have components satisfying the given equation with partial derivatives (see [4,

7]). Inasmuch as monogenic functions taking values in a commutative Banach

algebra form a functional algebra, note that a relation between these functions and

solutions of given equations with partial derivatives is important for constructing

mentioned solutions.

1.6. Three-dimensional harmonic algebras

The problem on ﬁnding a three-dimensional harmonic algebra 𝔸with the unit 1

was completely solved by I.P. Mel’nichenko [3, 4, 7]. In the paper [7] I.P. Mel’nichen-

ko established that there does not exist a harmonic algebra of third rank with a

unit over the ﬁeld ℝ, but he constructed a three-dimensional harmonic algebra

over the ﬁeld ℂ.

Commutative Algebras and Equations of Mathematical Physics 181

Theorem 1.1 (I. Mel’nichenko, [7]). The commutative associative algebra 𝔸is har-

monic, if the multiplication table for the basis {𝑒1,𝑒

2,𝑒

3}is of the following form:

𝑒𝑘𝑒1=𝑒𝑘,𝑘=1,2,3;

𝑒2𝑒2=−1

2𝑒1−𝑖

2(sin 𝜔)𝑒2+𝑖

2(cos 𝜔)𝑒3,

𝑒2𝑒3=𝑖

2(cos 𝜔)𝑒2+𝑖

2(sin 𝜔)𝑒3,

𝑒3𝑒3=−1

2𝑒1+𝑖

2(sin 𝜔)𝑒2−𝑖

2(cos 𝜔)𝑒3,

(1.8)

where 𝑖is the imaginary complex unit and 𝜔∈ℂ.

Let {𝑒1,𝑒

2,𝑒

3}be a harmonic basis in the algebra 𝔸with the multiplication

table (1.8). Associate with a set 𝑄⊂ℝ3the set 𝑄𝜁:= {𝜁=𝑥𝑒1+𝑦𝑒2+𝑧𝑒3:

(𝑥, 𝑦, 𝑧 )∈𝑄}in 𝐸3.

Theorem 1.2 (I. Mel’nichenko, [7]). Let 𝔸be a harmonic algebra with the multi-

plication table (1.8) for a basis {𝑒1,𝑒

2,𝑒

3}.IfafunctionΦ:Ω

𝜁→𝔸is monogenic

in a domain Ω𝜁⊂𝐸3, then the components 𝑈𝑘:Ω→ℝ,𝑉𝑘:Ω→ℝ,𝑘=1,2,3,

of decomposition

Φ(𝑥𝑒1+𝑦𝑒2+𝑧𝑒3)=



𝑘=1

𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑒𝑘+𝑖



𝑘=1

𝑉𝑘(𝑥, 𝑦, 𝑧)𝑒𝑘,(𝑥, 𝑦, 𝑧 )∈Ω,

generate harmonic vectors V1:= (𝑈1,−1

2𝑈2,−1

2𝑈3),V2:= (𝑉1,−1

2𝑉2,−1

2𝑉3).

I. Mel’nichenko [3, 4] found all three-dimensional harmonic algebras and de-

veloped a method for ﬁnding all harmonic bases in these algebras.

Note that there exist only four commutative associative algebras of third

rank with unit 1 over the ﬁeld ℂ. If one chooses nilpotent and idempotent elements

generating these algebras, then the multiplication tables will be of the most simple

form.

Let 𝔸1be a semisimple algebra with idempotent elements in the basis

{ℐ1,ℐ2,ℐ3}and the multiplication table

ℐ2

1=𝐼1,ℐ2

2=ℐ2,ℐ2

3=ℐ3,ℐ1ℐ2=ℐ1ℐ3=ℐ2ℐ3=0.

Here 1 = ℐ1+ℐ2+ℐ3.

Other algebras contain radicals.

Let 𝔸2be an algebra with basis {ℐ1,ℐ2,𝜌}and multiplication table

ℐ2

1=ℐ1,ℐ2

2=ℐ2,ℐ1ℐ2=0,𝜌

2=0,ℐ1𝜌=0,ℐ2𝜌=𝜌.

Here 1 = ℐ1+ℐ2,and 𝜌is a radical of the algebra.

Algebras 𝔸3and 𝔸4with the basis {1,𝜌

1,𝜌

2}have no ideals generated by

idempotents and 𝜌1and 𝜌2are radicals of these algebras.

182 S.A. Plaksa

The multiplication table in the algebra 𝔸3is of the form

𝜌2

1=𝜌2,𝜌

2=0,𝜌

1𝜌2=0.

The multiplication table in the algebra 𝔸4is of the form

𝜌2

1=𝜌2

2=𝜌1𝜌2=0.

Theorem 1.3 (I. Mel’nichenko, [3]). The algebra 𝔸4is not harmonic. The algebras

𝔸1,𝔸2,𝔸3are harmonic.

All harmonic bases in the algebras 𝔸1,𝔸2,𝔸3are described (see [4]). Note

that in the semisimple algebra 𝔸1, in particular, there exists the family of harmonic

bases constructed in Theorem 1.1.

In the algebra 𝔸2there exists the following family of harmonic bases:

𝑒1=1,

𝑒2=𝑖(sin 𝜔)ℐ1+𝑖(cos 𝜔)ℐ2−(sin 𝜔)𝜌,

𝑒3=𝑖(cos 𝜔)ℐ1−𝑖(sin 𝜔)ℐ2−(cos 𝜔)𝜌,

and in the algebra 𝔸3there exists the following family of harmonic bases:

𝑒1=1,

𝑒2=𝑖sin 𝜔+(cos𝜔)𝜌1+𝑖cos 𝜋

6−𝜔𝜌2,

𝑒3=𝑖cos 𝜔−(sin 𝜔)𝜌1+𝑖sin 𝜋

6−𝜔𝜌2,

where 𝜔∈ℂ(see [3, 4]).

2. Algebraic-analytic properties of monogenic functions

in the algebra 𝔸3

2.1. Harmonic bases in the algebra 𝔸3

All harmonic bases in the algebra 𝔸3are described in Theorem 1.6 [4], videlicet,

the following statement is true:

Theorem 2.1. Abasis{𝑒1,𝑒

2,𝑒

3}is harmonic if decompositions of its elements

with re spect t o the basis {1,𝜌

1,𝜌

2}are of the form

𝑒1=1,

𝑒2=𝑛1+𝑛2𝜌1+𝑛3𝜌2,

𝑒3=𝑚1+𝑚2𝜌1+𝑚3𝜌2,

(2.1)

where 𝑛𝑘and 𝑚𝑘for 𝑘=1,2,3are complex numbers satisfying the system of

equat ions

1+𝑛2

1+𝑚2

1=0,

𝑛1𝑛2+𝑚1𝑚2=0,

𝑥𝑛2

2+𝑚2

2+2(𝑛1𝑛3+𝑚1𝑚3)=0

(2.2)

Commutative Algebras and Equations of Mathematical Physics 183

and the inequality 𝑛2𝑚3−𝑛3𝑚2∕=0,andmoreover,atleastoneofthenumbersin

each of the pairs (𝑛1,𝑛

2)and (𝑚1,𝑚

2)is not equal to zero. Any harmonic basis

in 𝔸3can be obtained as a result of multiplication of elements of harmonic basis

(2.1) by an invertible element of the algebra 𝔸3.

For example, if 𝑛1=𝑖,𝑛2=𝑖/2, 𝑛3=𝑚1=0,𝑚2=−1, 𝑚3=−√3𝑖/2,

then we have a harmonic basis {𝑒0

1,𝑒

2,𝑒

3}with the following decomposition with

respect to the basis {1,𝜌

1,𝜌

2}:

𝑒0

1=1,𝑒

2=𝑖+𝑖

2𝜌2,𝑒

3=−𝜌1−√3

2𝑖𝜌2.(2.3)

The algebra 𝔸3has the unique maximal ideal ℐ:= {𝜆1𝜌1+𝜆2𝜌2:𝜆1,𝜆

2∈ℂ}

which is also the radical of 𝔸3.

Consider the linear functional 𝑓:𝔸3→ℂsuch that the maximal ideal ℐis

its kernel and 𝑓(1) = 1. It is well known [8, p. 135] that 𝑓is also a multiplicative

functional, i.e., the equality 𝑓(𝑎𝑏)=𝑓(𝑎)𝑓(𝑏) is fulﬁlled for all 𝑎,𝑏∈𝔸3.

2.2. Cauchy–Riemann conditions for functions taking values in the algebra 𝔸3

Let {𝑒1,𝑒

2,𝑒

3}be a harmonic basis of the form (2.1).

Let Ω be a domain in ℝ3and 𝜁=𝑥+𝑦𝑒2+𝑧𝑒3,where(𝑥, 𝑦, 𝑧 )∈Ω. Consider

the decomposition

Φ(𝜁)=



𝑘=1

𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑒𝑘,(2.4)

of a function Φ : Ω𝜁−→ 𝔸3with respect to the basis {𝑒1,𝑒

2,𝑒

3},wherethe

functions 𝑈𝑘:Ω−→ ℂare diﬀerentiable in Ω, i.e.,

𝑈𝑘(𝑥+Δ𝑥, 𝑦 +Δ𝑦, 𝑧 +Δ𝑧)−𝑈𝑘(𝑥, 𝑦 , 𝑧)

=∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑥 Δ𝑥+∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑦 Δ𝑦+∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑧 Δ𝑧

+𝑜(Δ𝑥)2+(Δ𝑦)2+(Δ𝑧)2,(Δ𝑥)2+(Δ𝑦)2+(Δ𝑧)2→0.

It follows from Theorem 1.3 [4] that the function Φ is monogenic in the domain

Ω𝜁if and only if the following Cauchy–Riemann conditions are satisﬁed in Ω𝜁:

∂Φ

∂𝑦 =∂Φ

∂𝑥 𝑒2,∂Φ

∂𝑧 =∂Φ

∂𝑥 𝑒3.(2.5)

2.3. A constructive description of monogenic functions taking values

in the algebra 𝔸3

Below in Section 2, all stated results are obtained jointly with V.S. Shpakivskyi

(see also [9]).

Let {𝑒1,𝑒

2,𝑒

3}be a harmonic basis of the form (2.1) and 𝜁=𝑥+𝑦𝑒2+𝑧𝑒3,

where 𝑥, 𝑦, 𝑧 ∈ℝ.

184 S.A. Plaksa

It follows from the equality

(𝑡−𝜁)−1=1

𝑡−𝑥−𝑛1𝑦−𝑚1𝑧+𝑛2𝑦+𝑚2𝑧

(𝑡−𝑥−𝑛1𝑦−𝑚1𝑧)2𝜌1

+𝑛3𝑦+𝑚3𝑧

(𝑡−𝑥−𝑛1𝑦−𝑚1𝑧)2+(𝑛2𝑦+𝑚2𝑧)2

(𝑡−𝑥−𝑛1𝑦−𝑚1𝑧)3𝜌2

∀𝑡∈ℂ:𝑡∕=𝑥+𝑛1𝑦+𝑚1𝑧(2.6)

(see [4, p. 30]) that the element 𝜁=𝑥+𝑦𝑒2+𝑧𝑒3∈𝐸3is noninvertible in 𝔸3if

and only if the point (𝑥, 𝑦, 𝑧) belongs to the following straight line in ℝ3:

𝐿:𝑥+𝑦ℜ𝑛1+𝑧ℜ𝑚1=0,

𝑦ℑ𝑛1+𝑧ℑ𝑚1=0.

We say that the domain Ω ⊂ℝ3is convex in the direction of the straight line

𝐿if Ω contains every segment parallel to 𝐿and connecting two points (𝑥1,𝑦

1,𝑧

1),

(𝑥2,𝑦

2,𝑧

2)∈Ω.

To obtain a constructive description of monogenic functions given in the

domain Ω𝜁and taking values in the algebra 𝔸3, consider an auxiliary statement.

Lemma 2.2. Let a domain Ω⊂ℝ3be convex in the direction of the straight line

𝐿and Φ:Ω

𝜁→𝔸3be a monogenic function in the domain Ω𝜁.If𝜁1,𝜁

2∈Ω𝜁and

𝜁2−𝜁1∈𝐿𝜁,then

Φ(𝜁1)−Φ(𝜁2)∈ℐ.(2.7)

Proof. Let the segment connecting the points (𝑥1,𝑦

1,𝑧

1),(𝑥2,𝑦

2,𝑧

2)∈Ωbepar-

allel to the straight line 𝐿.

Let us construct in Ω two surfaces 𝑄and Σ satisfying the following conditions:

∙𝑄and Σ have the same edge;

∙the surface 𝑄contains the point (𝑥1,𝑦

1,𝑧

1) and the surface Σ contains the

point (𝑥2,𝑦

2,𝑧

2);

∙restrictions of the functional 𝑓onto the sets 𝑄𝜁and Σ𝜁are one-to-one map-

pings of these sets onto the same domain 𝐺of the complex plane;

∙for every 𝜁0∈𝑄𝜁(and 𝜁0∈Σ𝜁) the equality

lim

𝜀→0+0 (Φ(𝜁0+𝜀(𝜁−𝜁0)) −Φ(𝜁0)) 𝜀−1=Φ

′(𝜁0)(𝜁−𝜁0) (2.8)

is fulﬁlled for all 𝜁∈𝑄𝜁for which 𝜁0+𝜀(𝜁−𝜁0)∈𝑄𝜁for all 𝜀∈(0,1) (or

for all 𝜁∈Σ𝜁for which 𝜁0+𝜀(𝜁−𝜁0)∈Σ𝜁for all 𝜀∈(0,1), respectively).

As the surface 𝑄, we can take an equilateral triangle having the center

(𝑥1,𝑦

1,𝑧

1) and apexes 𝐴1,𝐴

2,𝐴

3, and, in addition, the plane of this triangle is

perpendicular to the straight line 𝐿.

To construct the surface Σ, ﬁrst, consider a triangle with the center (𝑥2,𝑦

2,𝑧

and apexes 𝐴′

1,𝐴

′

2,𝐴

′

3such that the segments 𝐴′

1𝐴′

2,𝐴′

2𝐴′

3,𝐴′

1𝐴′

3are parallel to

the segments 𝐴1𝐴2,𝐴2𝐴3,𝐴1𝐴3, respectively, and, in addition, the length of

𝐴′

1𝐴′

2is 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

′′

3is

Commutative Algebras and Equations of Mathematical Physics 185

completely contained in Ω, where the points 𝐴′′

1,𝐴

′′

2,𝐴

′′

3are located in the plane

of triangle 𝐴1𝐴2𝐴3and the edges 𝐴′

𝑚𝐴′′

𝑚are parallel to 𝐿for 𝑚= 1,3.

Further, set a triangle with apexes 𝐵1,𝐵

2,𝐵

3such that the point 𝐵𝑚is

located on the segment 𝐴′

𝑚𝐴′′

𝑚for 𝑚= 1,3 and the truncated pyramid with

vertexes 𝐴1,𝐴

2,𝐴

3,𝐵

1,𝐵

2,𝐵

3and lateral edges 𝐴𝑚𝐵𝑚,𝑚= 1,3, is completely

contained in the domain Ω.

At last, in the plane of triangle 𝐴′

1𝐴′

2𝐴′

3set a triangle 𝑇with apexes 𝐶1,𝐶

2,𝐶

such that the segments 𝐶1𝐶2,𝐶2𝐶3,𝐶1𝐶3are parallel to the segments 𝐴′

1𝐴′

𝐴′

2𝐴′

3,𝐴′

1𝐴′

3, respectively, and, in addition, the length of 𝐶1𝐶2is less than the

length of 𝐴′

1𝐴′

2. It is evident that the truncated pyramid with vertexes 𝐵1,𝐵2,

𝐵3,𝐶1,𝐶2,𝐶3and 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 𝑇and the

lateral surfaces of mentioned truncated pyramids

𝐴1𝐴2𝐴3𝐵1𝐵2𝐵3and 𝐵1𝐵2𝐵3𝐶1𝐶2𝐶3.

For each 𝜉∈𝐺deﬁne two complex-valued functions 𝐻1and 𝐻2so that

𝐻1(𝜉):=𝑓(Φ(𝜁)),where 𝜉=𝑓(𝜁)and𝜁∈𝑄𝜁,

𝐻2(𝜉):=𝑓(Φ(𝜁)),where 𝜉=𝑓(𝜁)and𝜁∈Σ𝜁.

Inasmuch as 𝑓is a linear continuous multiplicative functional, from the equal-

ity (2.8) it follows that

lim

𝜀→0+0 (𝑓(Φ(𝜁0+𝜀(𝜁−𝜁0))) −𝑓(Φ(𝜉))) 𝜀−1=𝑓(Φ′(𝜁0))(𝑓(𝜁)−𝑓(𝜁0)).

Thus, there exist all directional derivatives of the functions 𝐻1,𝐻

2in the point

𝑓(𝜁0)∈𝐺, and, moreover, these derivatives are equal for each of the functions

𝐻1,𝐻

2. Therefore, by Theorem 21 in [10], the functions 𝐻1,𝐻

2are analytic in the

domain 𝐺, i.e., they are holomorphic in the case where 𝜉=𝜏+𝑖𝜂,andtheyare

antiholomorphic in the case where 𝜉=𝜏−𝑖𝜂,𝜏, 𝜂 ∈ℝ.

Inasmuch as 𝐻1(𝜉)≡𝐻2(𝜉) on the boundary of domain 𝐺, this identity is

fulﬁlled everywhere in 𝐺. Therefore, the equalities

𝑓(Φ(𝜁2)−Φ(𝜁1)) = 𝑓(Φ(𝜁2)) −𝑓(Φ(𝜁1)) = 0,

are fulﬁlled for 𝜁1:= 𝑥1+𝑦1𝑒2+𝑧1𝑒3and 𝜁2:= 𝑥2+𝑦2𝑒2+𝑧2𝑒3.Thus,Φ(𝜁2)−Φ(𝜁1)

belongs to the kernel ℐof functional 𝑓. The lemma is proved. □

Let 𝐷:= 𝑓(Ω𝜁)and𝐴be the linear operator which assigns the function

𝐹:𝐷→ℂto every monogenic function Φ : Ω𝜁→𝔸3by the formula 𝐹(𝜉):=

𝑓(Φ(𝜁)), where 𝜁=𝑥𝑒1+𝑦𝑒2+𝑧𝑒3and 𝜉:= 𝑓(𝜁)=𝑥+𝑛1𝑦+𝑚1𝑧. It follows

from Lemma 2.2 that the value 𝐹(𝜉) does not depend on a choice of a point 𝜁,for

which 𝑓(𝜁)=𝜉.

186 S.A. Plaksa

Theorem 2.3. If a domain Ω⊂ℝ3is convex in the direction of the straight line

𝐿, then every monogenic function Φ:Ω

𝜁→𝔸3canbeexpressedintheform

Φ(𝜁)= 1

2𝜋𝑖 

Γ𝜁

(𝐴Φ)(𝑡)(𝑡−𝜁)−1𝑑𝑡 +Φ

0(𝜁)∀𝜁∈Ω𝜁,(2.9)

where Γ𝜁is an arbitrary closed Jordan rectiﬁable curve in 𝐷that is homotopic to

the point 𝑓(𝜁)and embraces this point, and Φ0:Ω

𝜁→ℐis a monogenic function

taking values in the radical ℐ.

Proof. It is easy to see that the function Φ0from (2.9) belongs to the kernel of

the operator 𝐴, i.e., Φ0(𝜁)∈ℐfor all 𝜁∈Ω𝜁. The theorem is proved. □

Note that the complex number 𝜉=𝑓(𝜁) is the spectrum of 𝜁∈𝔸3,andthe

integral in the equality (2.9) is the principal extension (see [8, p. 165]) of analytic

function 𝐹(𝜉)=(𝐴Φ)(𝜉) of the complex variable 𝜉into the domain Ω𝜁.

It follows from Theorem 2.3 that the algebra of monogenic in Ω𝜁functions is

decomposed into the direct sum of the algebra of principal extensions of analytic

functions of the complex variable and the algebra of monogenic in Ω𝜁functions

taking values in the radical ℐ.

In Theorem 1.7 in [4] the principal extension of analytic function 𝐹:𝐷→ℂ

into the domain Π𝜁:= {𝜁∈𝐸3:𝑓(𝜁)∈𝐷}was explicitly constructed in the form

2𝜋𝑖 

Γ𝜁

𝐹(𝑡)(𝑡−𝜁)−1𝑑𝑡 =𝐹(𝑥+𝑛1𝑦+𝑚1𝑧)+(𝑛2𝑦+𝑚2𝑧)𝐹′(𝑥+𝑛1𝑦+𝑚1𝑧)𝜌1

+(𝑛3𝑦+𝑚3𝑧)𝐹′(𝑥+𝑛1𝑦+𝑚1𝑧)+ (𝑛2𝑦+𝑚2𝑧)2

2𝐹′′(𝑥+𝑛1𝑦+𝑚1𝑧)𝜌2

∀𝜁=𝑥𝑒1+𝑦𝑒2+𝑧𝑒3∈Π𝜁.(2.10)

It is evident that the domain Π ⊂ℝ3congruent to Π𝜁is an inﬁnite cylinder, and

its generatrix is parallel to 𝐿.

In the following theorem we describe all monogenic functions given in the

domain Ω𝜁and taking values in the radical ℐ.

Theorem 2.4. If a domain Ω⊂ℝ3is convex in the direction of the straight line

𝐿, then every monogenic function Φ0:Ω

𝜁→ℐcan be expressed in the form

Φ0(𝜁)=𝐹1(𝜉)𝜌1+(𝐹2(𝜉)+(𝑛2𝑦+𝑚2𝑧)𝐹′

1(𝜉)) 𝜌2

∀𝜁=𝑥𝑒1+𝑦𝑒2+𝑧𝑒3∈Ω𝜁,(2.11)

where 𝐹1,𝐹2are complex-valued analytic functions in the domain 𝐷and 𝜉=

𝑥+𝑛1𝑦+𝑚1𝑧.

Proof. For a monogenic function Φ0of the form

Φ0(𝜁)=𝑉1(𝑥, 𝑦, 𝑧)𝜌1+𝑉2(𝑥, 𝑦, 𝑧 )𝜌2,(2.12)

Commutative Algebras and Equations of Mathematical Physics 187

where 𝑉𝑘:Ω→ℂfor 𝑘=1,2, the Cauchy–Riemann conditions (2.5) are satisﬁed

with Φ = Φ0.

Substituting the expressions (2.1), (2.12) into the equalities (2.5) and taking

into account the uniqueness of decomposition of element of 𝔸3with respect to the

basis {1,𝜌

1,𝜌

2}, we get the following system for the determination of functions

𝑉1,𝑉

∂𝑉1

∂𝑦 =𝑛1

∂𝑉1

∂𝑥 ,

∂𝑉2

∂𝑦 =𝑛2

∂𝑉1

∂𝑥 +𝑛1

∂𝑉2

∂𝑥 ,

∂𝑉1

∂𝑧 =𝑚1

∂𝑉1

∂𝑥 ,

∂𝑉2

∂𝑧 =𝑚2

∂𝑉1

∂𝑥 +𝑚1

∂𝑉2

∂𝑥 .

(2.13)

Inasmuch as

𝜉=(𝑥+𝑦ℜ𝑛1+𝑧ℜ𝑚1)+𝑖(𝑦ℑ𝑛1+𝑧ℑ𝑚1)=:𝜏+𝑖𝜂 , (2.14)

from the ﬁrst and the third equations of the system (2.13) we get

∂𝑉1

∂𝜂 ℑ𝑛1=𝑖∂𝑉1

∂𝜏 ℑ𝑛1,∂𝑉1

∂𝜂 ℑ𝑚1=𝑖∂𝑉1

∂𝜏 ℑ𝑚1.(2.15)

It follows from the ﬁrst equation of the system (2.2) that, at least one of

the numbers Im 𝑛1,Im𝑚1is not equal to zero. Therefore, from (2.15) we get the

equality

∂𝑉1

∂𝜂 =𝑖∂𝑉1

∂𝜏 .(2.16)

Let us prove that 𝑉1(𝑥1,𝑦

1,𝑧

1)=𝑉1(𝑥2,𝑦

2,𝑧

2) for the points (𝑥1,𝑦

1,𝑧

1),

(𝑥2,𝑦

2,𝑧

2)∈Ω such that the segment connecting these points is parallel to the

straight line 𝐿. Consider two surfaces 𝑄, Σ in Ω and the domain 𝐺in ℂthat are

deﬁned in the proof of Lemma 2.2. For each 𝜉∈𝐺deﬁne two complex-valued

functions 𝐻1and 𝐻2so that

𝐻1(𝜉):=𝑉1(𝑥, 𝑦, 𝑧)for(𝑥, 𝑦, 𝑧 )∈𝑄,

𝐻2(𝜉):=𝑉1(𝑥, 𝑦, 𝑧)for(𝑥, 𝑦, 𝑧 )∈Σ,

where the correspondence between the points (𝑥, 𝑦, 𝑧)and𝜉∈𝐺is determined by

the relation (2.14). The functions 𝐻1,𝐻

2are analytic in the domain 𝐺owing to

the equality (2.16) and Theorem 6 in [11]. Further, the identity 𝐻1(𝜉)≡𝐻2(𝜉)in

𝐺can be proved in the same way as in the proof of Lemma 2.2.

Thus, the function 𝑉1of the form 𝑉1(𝑥, 𝑦, 𝑧 ):=𝐹1(𝜉), where 𝐹1(𝜉)isan

arbitrary function analytic in 𝐷, is the general solution of the system consisting

of the ﬁrst and the third equations of the system (2.13).

188 S.A. Plaksa

Now, from the second and the fourth equations of the system (2.13) we get

the following system for the determination of function 𝑉2(𝑥, 𝑦, 𝑧):

∂𝑉2

∂𝑦 −𝑛1

∂𝑉2

∂𝑥 =𝑛2

∂𝐹1

∂𝑥 ,

∂𝑉2

∂𝑧 −𝑚1

∂𝑉2

∂𝑥 =𝑚2

∂𝐹1

∂𝑥 .

(2.17)

The function (𝑛2𝑦+𝑚2𝑧)𝐹′

1(𝜉) is a particular solution of this system and,

therefore, the general solution of the system (2.17) is represented in the form

𝑉2(𝑥, 𝑦, 𝑧)=𝐹2(𝜉)+(𝑛2𝑦+𝑚2𝑧)𝐹′

1(𝜉),

where 𝐹2is an arbitrary function analytic in 𝐷. The theorem is proved. □

It follows from the equalities (2.9), (2.11) that in the case where a domain

Ω⊂ℝ3is convex in the direction of the straight line 𝐿, any monogenic function

Φ:Ω

𝜁→𝔸3can be constructed by means of three complex analytic in 𝐷functions

𝐹,𝐹1,𝐹2in the form:

Φ(𝜁)= 1

2𝜋𝑖 

Γ𝜁

𝐹(𝑡)(𝑡−𝜁)−1𝑑𝑡 +𝜌1𝐹1(𝑥+𝑛1𝑦+𝑚1𝑧)

+𝜌2𝐹2(𝑥+𝑛1𝑦+𝑚1𝑧)+(𝑛2𝑦+𝑚2𝑧)𝐹′

1(𝑥+𝑛1𝑦+𝑚1𝑧)

∀𝜁=𝑥𝑒1+𝑦𝑒2+𝑧𝑒3∈Ω𝜁,(2.18)

and in this case the equality (2.10) is applicable. Thus, we can rewrite the equality

(2.18) in the following form:

Φ(𝜁)=𝐹(𝑥+𝑛1𝑦+𝑚1𝑧)

+𝐹1(𝑥+𝑛1𝑦+𝑚1𝑧)+(𝑛2𝑦+𝑚2𝑧)𝐹′(𝑥+𝑛1𝑦+𝑚1𝑧)𝜌1

+𝐹2(𝑥+𝑛1𝑦+𝑚1𝑧)+(𝑛2𝑦+𝑚2𝑧)𝐹′

1(𝑥+𝑛1𝑦+𝑚1𝑧)

+(𝑛3𝑦+𝑚3𝑧)𝐹′(𝑥+𝑛1𝑦+𝑚1𝑧)+ (𝑛2𝑦+𝑚2𝑧)2

2𝐹′′(𝑥+𝑛1𝑦+𝑚1𝑧)𝜌2

∀𝜁=𝑥𝑒1+𝑦𝑒2+𝑧𝑒3∈Ω𝜁.(2.19)

Using the equality (2.19), one can construct all monogenic functions Φ : Ω𝜁→𝔸3

by means of arithmetic operations with arbitrary complex-valued analytic func-

tions 𝐹,𝐹1,𝐹2giveninthedomain𝐷⊂ℂ.

It is evident that the following statement follows from the equality (2.19).

Theorem 2.5. If a domain Ω⊂ℝ3is convex in the direction of the straight line

𝐿, then every monogenic function Φ:Ω

𝜁→𝔸3can be continued t o a function

monogenic in the domain Π𝜁.

Commutative Algebras and Equations of Mathematical Physics 189

Note that the condition of convexity of Ω in the direction of the line 𝐿is

essential for the truth of Lemma 2.2 and consequently for the truth of Theorems

2.3–2.5.

Example 2.6. Let us construct a domain Ω, which is not convex in the direction of

the straight line 𝐿, and an example of monogenic function Φ : Ω𝜁→𝔸3for which

the relation (2.7) is not fulﬁlled for certain 𝜁1,𝜁

2∈Ω𝜁such that 𝜁2−𝜁1∈𝐿𝜁.

Consider a harmonic basis (2.3). In this case the straight line 𝐿coincide with

the axis 𝑂𝑧. Consider the domain Ω𝜁which is the union of sets

Ω(1)

𝜁:={𝑥+𝑦𝑒0

2+𝑧𝑒0

3∈𝐸3:∣𝑥+𝑖𝑦∣<2,0<𝑧<2,−𝜋/4<arg(𝑥+𝑖𝑦)<3𝜋/2},

Ω(2)

𝜁:={𝑥+𝑦𝑒0

2+𝑧𝑒0

3∈𝐸3:∣𝑥+𝑖𝑦∣<2,2≤𝑧≤4,𝜋/2<arg(𝑥+𝑖𝑦)<3𝜋/2},

Ω(3)

𝜁:={𝑥+𝑦𝑒0

2+𝑧𝑒0

3∈𝐸3:∣𝑥+𝑖𝑦∣<2,4<𝑧<6,𝜋/2<arg(𝑥+𝑖𝑦)<9𝜋/4}.

It is evident that the domain Ω ⊂ℝ3congruent to Ω𝜁is not convex in the direction

of the axis 𝑂𝑧.

In the domain {𝜉∈ℂ:∣𝜉∣<2,−𝜋/4<arg 𝜉<3𝜋/2}of the complex

plane consider a holomorphic branch 𝐻1(𝜉) of analytic function Ln 𝜉for which

𝐻1(1) = 0. In the domain {𝜉∈ℂ:∣𝜉∣<2,𝜋/2<arg 𝜉<9𝜋/4}consider also a

holomorphic branch 𝐻2(𝜉) of function Ln 𝜉for which 𝐻2(1) = 2𝜋𝑖.

Further, consider the principal extension Φ1of function 𝐻1into the set

Ω(1)

𝜁∪Ω(2)

𝜁and the principal extension Φ2of function 𝐻2into the set Ω(2)

𝜁∪Ω(3)

𝜁

constructed by using the formula (2.10):

Φ1(𝜁)=𝐻1(𝑥+𝑖𝑦)−2𝑧−𝑖𝑦

2(𝑥+𝑖𝑦)𝜌1−√3𝑖𝑧

2(𝑥+𝑖𝑦)+(2𝑧−𝑖𝑦)2

8(𝑥+𝑖𝑦)2𝜌2,

Φ2(𝜁)=𝐻2(𝑥+𝑖𝑦)−2𝑧−𝑖𝑦

2(𝑥+𝑖𝑦)𝜌1−√3𝑖𝑧

2(𝑥+𝑖𝑦)+(2𝑧−𝑖𝑦)2

8(𝑥+𝑖𝑦)2𝜌2,

where 𝜁=𝑥+𝑦𝑒0

2+𝑧𝑒0

Now, the function

Φ(𝜁):=Φ1(𝜁)for𝜁∈Ω(1)

𝜁∪Ω(2)

𝜁,

Φ2(𝜁)for𝜁∈Ω(3)

𝜁

is monogenic in the domain Ω𝜁, because Φ1(𝜁)≡Φ2(𝜁)everywhereinΩ

(2)

𝜁.Atthe

same time, for the points 𝜁1=1+𝑒0

3and 𝜁2=1+5𝑒0

3we have 𝜁2−𝜁1∈𝐿𝜁but

Φ(𝜁2)−Φ(𝜁1)=2𝜋𝑖 −4𝜌1−(12+2

√3𝑖)𝜌2/∈ℐ,

i.e., the relation (2.7) is not fulﬁlled.

190 S.A. Plaksa

Taking into account the equality (2.10), we can rewrite the equality (2.19) in

the following integral form:

Φ(𝜁)= 1

2𝜋𝑖 

Γ𝜁𝐹(𝑡)+𝜌1𝐹1(𝑡)+𝜌2𝐹2(𝑡)(𝑡−𝜁)−1𝑑𝑡 ∀𝜁∈Ω𝜁,(2.20)

where the curve Γ𝜁is the same as in Theorem 2.3.

The following statement is true for monogenic functions in an arbitrary do-

main Ω𝜁.

Theorem 2.7. For every monogenic function Φ:Ω

𝜁→𝔸3in an arbitrary domain

Ω𝜁, the Gateaux 𝑛th derivatives Φ(𝑛)are monogenic functions in Ω𝜁for any 𝑛.

Proof. Consider an arbitrary point (𝑥0,𝑦

0,𝑧

0)∈Ωandaball℧⊂Ωwiththe

center in the point (𝑥0,𝑦

0,𝑧

0). Inasmuch as ℧is a convex set, in the neighbourhood

℧𝜁of the point 𝜁0=𝑥0+𝑦0𝑒2+𝑧0𝑒3we have the equality (2.20), where the integral

has the Gateaux 𝑛th derivative for any 𝑛and these derivatives are continuous

functions in ℧𝜁.Thus,theGateaux𝑛th derivative Φ(𝑛)is a monogenic function in

℧𝜁for any 𝑛. The theorem is proved. □

Using the integral expression (2.20) of monogenic function Φ : Ω𝜁→𝔸3,we

obtain the following expression for the Gateaux 𝑛th derivative Φ(𝑛):

Φ(𝑛)(𝜁)= 𝑛!

2𝜋𝑖 

Γ𝜁𝐹(𝑡)+𝜌1𝐹1(𝑡)+𝜌2𝐹2(𝑡)(𝑡−𝜁)−1𝑛+1𝑑𝑡 ∀𝜁∈Ω𝜁.

2.4. Isomorphism of algebras of monogenic functions

Let us establish an isomorphism between algebras of monogenic functions at tran-

sition from a harmonic basis to another one. By ℳ(𝐸3,Ω𝜁) we denote the algebra

of monogenic functions in a domain Ω𝜁⊂𝐸3.

Consider a harmonic basis (2.3) and 𝐸0

3:= {𝜁=𝑥𝑒0

1+𝑦𝑒0

2+𝑧𝑒0

3:𝑥, 𝑦, 𝑧 ∈ℝ}.

Let Ω0

𝜁be a domain in 𝐸0

3. Consider also an arbitrary harmonic basis {˜𝑒1,˜𝑒2,˜𝑒3}

in 𝔸3and 

𝐸3:= {

𝜁=˜𝑥˜𝑒1+˜𝑦˜𝑒2+˜𝑧˜𝑒3:˜𝑥, ˜𝑦, ˜𝑧∈ℝ}.

Let us specify a correspondence between 𝐸0

3and 

𝐸3under which a domain



Ω˜

𝜁⊂

𝐸3corresponds to the domain Ω0

𝜁and algebras ℳ(𝐸0

3,Ω0

𝜁), ℳ(

𝐸3,

Ω˜

𝜁)are

isomorphic.

It follows from Theorem 2.1 that elements of the basis {˜𝑒1,˜𝑒2,˜𝑒3}can be

represented in the form ˜𝑒1=𝑎𝑒1,˜𝑒2=𝑎𝑒2,˜𝑒3=𝑎𝑒3,where 𝑎is an invert-

ible element and decompositions of the basis {𝑒1,𝑒

2,𝑒

3}with respect to the basis

{1,𝜌

1,𝜌

2}are of the form (2.1), where without restricting the generality we may

assume that Im 𝑛1∕= 0. Then the basis {𝑒1,𝑒

2,𝑒

3}can be expressed in the form

𝑒1=𝑒0

𝑒2=𝛼1𝑒0

1+𝛼2𝑒0

2+𝑟21𝜌1+𝑟22 𝜌2,

𝑒3=𝛽1𝑒0

1+𝛽2𝑒0

2+𝑒0

3+𝑟31𝜌1+𝑟32 𝜌2,

Commutative Algebras and Equations of Mathematical Physics 191

where 𝛼1:= Re 𝑛1,𝛼

2:= Im 𝑛1∕=0,𝛽

1:= Re 𝑚1,𝛽

2:= Im 𝑚1,𝑟

21 := 𝑛2,

𝑟22 := 𝑛3−1

2𝑖Im 𝑛1,𝑟

31 := 𝑚2+1,𝑟

32 := 𝑚3+√3

2𝑖−1

2𝑖Im 𝑚1.

Theorem 2.8. Let a corresponde nce bet ween 𝜁=𝑥𝑒0

1+𝑦𝑒0

2+𝑧𝑒0

3∈𝐸0

3and



𝜁=˜𝑥˜𝑒1+˜𝑦˜𝑒2+˜𝑧˜𝑒3∈

𝐸3be given by the equalities

𝑥=˜𝑥+𝛼1˜𝑦+𝛽1˜𝑧,

𝑦=𝛼2˜𝑦+𝛽2˜𝑧,

𝑧=˜𝑧.

Then the algebras ℳ(𝐸0

3,Ω0

𝜁),ℳ(

𝐸3,

Ω˜

𝜁)are isomorphic, and the correspondence

ℳ(𝐸0

3,Ω0

𝜁)∋Φ←→ 

Φ∈ℳ(

𝐸3,

Ω˜

𝜁)are established by the equality



Φ(

𝜁)=Φ(𝜁)+Φ

′(𝜁)(𝑟21 ˜𝑦+𝑟31 ˜𝑧)𝜌1+(𝑟22 ˜𝑦+𝑟32 ˜𝑧)𝜌2+1

2Φ′′(𝜁)(𝑟21 ˜𝑦+𝑟31 ˜𝑧)2𝜌2.

3. Integral theorems in the algebra 𝔸3

3.1. On integral theorems in hypercomplex analysis

In the paper [12] for functions diﬀerentiable in the sense of Lorch in an arbitrary

convex domain of a commutative associative Banach algebra, some properties sim-

ilar to properties of holomorphic functions of a complex variable (in particular, the

integral Cauchy theorem and the integral Cauchy formula, the Taylor expansion

and the Morera theorem) are established. The convexity of the domain in the

mentioned results from [12] is excluded by E.K. Blum [13].

In this paper we establish similar results for monogenic functions Φ : Ω𝜁→𝔸3

given only in a domain Ω𝜁of the linear envelope 𝐸3instead of the domain of the

whole algebra 𝔸3. Let us note that apriori the diﬀerentiability of the function

Φ in the sense of Gateaux is a restriction weaker than the diﬀerentiability of this

function in the sense of Lorch. Moreover, note that the integral Cauchy formula

established in the papers [12, 13] is not applicable to a monogenic function Φ :

Ω𝜁→𝔸3because it deals with an integration along a curve on which the function

Φ is not given, generally speaking.

Note that as well as in [12, 13], some hypercomplex analogues of the integral

Cauchy theorem for a curvilinear integral are established in the papers [14, 15]. In

the papers [14, 16–18] similar theorems are established for surface integrals.

Below in Section 3, all stated results are obtained jointly with V.S. Sh-

pakivskyi (see also [19, 20]).

3.2. Cauchy integral theorem for a surface integral

Let {𝑒1=1,𝑒

2,𝑒

3}be a harmonic basis in the algebra 𝔸3.

Along with monogenic functions satisfying the Cauchy–Riemann conditions

(2.5), consider a function Ψ : Ω𝜁→𝔸3having continuous partial derivatives of

192 S.A. Plaksa

the ﬁrst order in a domain Ω𝜁and satisfying the equation

∂Ψ

∂𝑥 +∂Ψ

∂𝑦 𝑒2+∂Ψ

∂𝑧 𝑒3= 0 (3.1)

in every point of this domain.

In the scientiﬁc literature, diﬀerent denominations are used for functions

satisfying equations of the form (3.1). For example, in the papers [14, 15, 21] they

are called regular functions, and in the papers [16, 17, 22] they are called monogenic

functions. As well as in the papers [18, 23, 24], we call Ψ a hyperholomorphic

function if it satisﬁes equation (3.1).

It is well known that in the quaternion analysis the classes of functions de-

termined by means of conditions of the form (2.5) and (3.1) do not coincide (see

[14, 25]).

Note that in the algebra 𝔸3the set of monogenic functions is a subset of the

set of hyperholomorphic functions, because every monogenic function Φ : Ω𝜁→

𝔸3satisﬁes the equality (3.1) owing to conditions (1.5), (2.5). But, there exist

hyperholomorphic functions which are not monogenic. For example, the function

Ψ(𝑥+𝑦𝑒2+𝑧𝑒3)=𝑧𝑒2−𝑦𝑒3satisﬁes condition (3.1), but it does not satisfy

equalities of the form (2.5).

Let Ω be a bounded closed set in ℝ3. For a continuous function Ψ : Ω𝜁→𝔸3

of the form

Ψ(𝑥+𝑦𝑒2+𝑧𝑒3)=



𝑘=1

𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑒𝑘+𝑖



𝑘=1

𝑉𝑘(𝑥, 𝑦, 𝑧)𝑒𝑘,(3.2)

where (𝑥, 𝑦, 𝑧 )∈Ωand𝑈𝑘:Ω→ℝ,𝑉𝑘:Ω→ℝ, we deﬁne a volume integral by

the equality



Ω𝜁

Ψ(𝜁)𝑑𝑥𝑑𝑦𝑑𝑧 :=



𝑘=1

𝑒𝑘

𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑑𝑥𝑑𝑦𝑑𝑧 +𝑖



𝑘=1

𝑒𝑘

𝑉𝑘(𝑥, 𝑦, 𝑧)𝑑𝑥𝑑𝑦𝑑𝑧.

Let Σ be a quadrable surface in ℝ3with quadrable projections on the coor-

dinate planes. For a continuous function Ψ : Σ𝜁→𝔸3of the form (3.2), where

(𝑥, 𝑦, 𝑧 )∈Σand𝑈𝑘:Σ→ℝ,𝑉𝑘:Σ→ℝ, we deﬁne a surface integral on

Σ𝜁with the diﬀerential form 𝜎𝛼1,𝛼2,𝛼3:= 𝛼1𝑑𝑦𝑑𝑧 +𝛼2𝑑𝑧𝑑𝑥𝑒2+𝛼3𝑑𝑥𝑑𝑦𝑒3,where

𝛼1,𝛼

2,𝛼

3∈ℝ, by the equality



Σ𝜁

Ψ(𝜁)𝜎𝛼1,𝛼2,𝛼3:=



𝑘=1

𝑒𝑘

𝛼1𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑑𝑦𝑑𝑧 +



𝑘=1

𝑒2𝑒𝑘

𝛼2𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑑𝑧𝑑𝑥



𝑘=1

𝑒3𝑒𝑘

𝛼3𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑑𝑥𝑑𝑦 +𝑖



𝑘=1

𝑒𝑘

𝛼1𝑉𝑘(𝑥, 𝑦, 𝑧)𝑑𝑦𝑑𝑧

+𝑖



𝑘=1

𝑒2𝑒𝑘

𝛼2𝑉𝑘(𝑥, 𝑦, 𝑧)𝑑𝑧𝑑𝑥 +𝑖



𝑘=1

𝑒3𝑒𝑘

𝛼3𝑉𝑘(𝑥, 𝑦, 𝑧)𝑑𝑥𝑑𝑦.

Commutative Algebras and Equations of Mathematical Physics 193

A connected homeomorphic image of a square in ℝ3is called a simple surface.

Asurfaceislocally-simple if it is simple in a certain neighborhood of every point.

If a simply connected domain Ω ⊂ℝ3has a closed locally-simple piece-

smooth boundary ∂Ω and a function Ψ : Ω𝜁→𝔸3is continuous together with

partial derivatives of the ﬁrst order up to the boundary ∂Ω𝜁, then the following

analogue of the Gauss–Ostrogradsky formula is true:



∂Ω𝜁

Ψ(𝜁)𝜎=

Ω𝜁∂Ψ

∂𝑥 +∂Ψ

∂𝑦 𝑒2+∂Ψ

∂𝑧 𝑒3𝑑𝑥𝑑𝑦𝑑𝑧, (3.3)

where 𝜎:= 𝜎1,1,1≡𝑑𝑦𝑑𝑧 +𝑑𝑧𝑑𝑥𝑒2+𝑑𝑥𝑑𝑦𝑒3. Now, the next theorem is a result of

formula (3.3) and equality (3.1).

Theorem 3.1. Suppose that Ωis a simply connected domain with a closed locally-

simple piece-smooth boundary ∂Ω. Suppose also that the function Ψ:Ω𝜁→𝔸3is

continuous in the closure Ω𝜁of domain Ω𝜁and is hyperholomorphic in Ω𝜁.Then



∂Ω𝜁

Ψ(𝜁)𝜎=0.

3.3. Cauchy integral theorem for a curvilinear integral

Let 𝛾be a Jordan rectiﬁable curve in ℝ3. For a continuous function Ψ : 𝛾𝜁→𝔸3

of the form (3.2), where (𝑥, 𝑦, 𝑧)∈𝛾and 𝑈𝑘:𝛾→ℝ,𝑉𝑘:𝛾→ℝ, we deﬁne an

integral along the curve 𝛾𝜁by the equality



𝛾𝜁

Ψ(𝜁)𝑑𝜁 :=



𝑘=1

𝑒𝑘

𝛾

𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑑𝑥 +



𝑘=1

𝑒2𝑒𝑘

𝛾

𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑑𝑦



𝑘=1

𝑒3𝑒𝑘

𝛾

𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑑𝑧 +𝑖



𝑘=1

𝑒𝑘

𝛾

𝑉𝑘(𝑥, 𝑦, 𝑧)𝑑𝑥

+𝑖



𝑘=1

𝑒2𝑒𝑘

𝛾

𝑉𝑘(𝑥, 𝑦, 𝑧)𝑑𝑦 +𝑖



𝑘=1

𝑒3𝑒𝑘

𝛾

𝑉𝑘(𝑥, 𝑦, 𝑧)𝑑𝑧,

where 𝑑𝜁 := 𝑑𝑥 +𝑒2𝑑𝑦 +𝑒3𝑑𝑧.

If a function Φ : Ω𝜁→𝔸3is continuous together with partial derivatives of

the ﬁrst order in a domain Ω𝜁, and Σ is a piece-smooth surface in Ω, and the edge

𝛾of surface Σ is a rectiﬁable Jordan curve, then the following analogue of the

Stokes formula is true:



𝛾𝜁

Φ(𝜁)𝑑𝜁 =

Σ𝜁∂Φ

∂𝑥 𝑒2−∂Φ

∂𝑦 𝑑𝑥𝑑𝑦 +∂Φ

∂𝑦 𝑒3−∂Φ

∂𝑧 𝑒2𝑑𝑦𝑑𝑧

+∂Φ

∂𝑧 −∂Φ

∂𝑥 𝑒3𝑑𝑧𝑑𝑥.

(3.4)

194 S.A. Plaksa

Now, the next theorem is a result of formula (3.4) and equalities (2.5).

Theorem 3.2. Suppose that Φ:Ω

𝜁→𝔸3is a monogenic function in a domain

Ω𝜁,andΣis a piece-wise smooth surface in Ω,andtheedge𝛾of surface Σis a

rectiﬁable Jordan curve. Then



𝛾𝜁

Φ(𝜁)𝑑𝜁 =0.(3.5)

Now, similarly to the proof of Theorem 3.2 in [13] we can prove the following

Theorem 3.3. Let Φ:Ω

𝜁→𝔸3be a monogenic function in a domain Ω𝜁.Then

for every closed Jordan rectiﬁable curve 𝛾homotopic to a point in Ω,theequality

(3.5) is true.

For functions taking values in the algebra 𝔸3, the following Morera theorem

can be established in the usual way.

Theorem 3.4. If a function Φ:Ω

𝜁→𝔸3is continuous in a domain Ω𝜁and satisﬁes

the equality 

∂△𝜁

Φ(𝜁)𝑑𝜁 = 0 (3.6)

for every triangle △𝜁such that △𝜁⊂Ω𝜁, then the function Φis monogenic in the

domain Ω𝜁.

3.4. Cauchy integral formula

Inasmuch as according to Theorem 2.8 there exists an isomorphism between alge-

bras of monogenic functions Φ(𝜁)ofthevariable𝜁=𝑥𝑒1+𝑦𝑒2+𝑧𝑒3determined

in various harmonic bases, it is enough to study properties of monogenic functions

given in domains of a linear envelope 𝐸3generated by one of its harmonic bases.

Thus, let us consider a basis (2.3) as the harmonic basis {𝑒1,𝑒

2,𝑒

3}.

It follows from equality (2.6) that

𝜁−1=1

𝑥+𝑖𝑦 +𝑧

(𝑥+𝑖𝑦)2𝜌1+𝑖

√3𝑧−𝑦

(𝑥+𝑖𝑦)2+𝑧2

(𝑥+𝑖𝑦)3𝜌2(3.7)

for all 𝜁=𝑥+𝑦𝑒2+𝑧𝑒3∈𝐸3∖{𝑧𝑒3:𝑧∈ℝ}. Thus, it is obvious that the straight

line {𝑧𝑒3:𝑧∈ℝ}is contained in the radical of the algebra 𝔸3.

Using equality (3.7), it is easy to calculate that



˜𝛾𝜁

𝜏−1𝑑𝜏 =2𝜋𝑖, (3.8)

where ˜𝛾𝜁:= {𝜏=𝑥+𝑦𝑒2:𝑥2+𝑦2=𝑅2}.

Commutative Algebras and Equations of Mathematical Physics 195

Theorem 3.5. Let Ωbe a domain convex in the direction of the axis 𝑂𝑧 and Φ:

Ω𝜁→𝔸3be a monogenic function in the domain Ω𝜁. Then for every point 𝜁0∈Ω𝜁

the following equality is true:

Φ(𝜁0)= 1

2𝜋𝑖 

𝛾𝜁

Φ(𝜁)(𝜁−𝜁0)−1𝑑𝜁, (3.9)

where 𝛾𝜁is an arbitrary closed Jordan rectiﬁable curve in Ω𝜁, which winds once

around the straight line {𝜁0+𝑧𝑒3:𝑧∈ℝ}and is homotopic to the point 𝜁0.

Proof. We represent the integral from the right-hand side of equality (3.9) as the

sum of the following two integrals:

𝛾𝜁

Φ(𝜁)(𝜁−𝜁0)−1𝑑𝜁 =𝛾𝜁

(Φ(𝜁)−Φ(𝜁0)) (𝜁−𝜁0)−1𝑑𝜁

+Φ(𝜁0)𝛾𝜁

(𝜁−𝜁0)−1𝑑𝜁 =: 𝐼1+𝐼2.

Inasmuch as the domain Ω is convex in the direction of the axis 𝑂𝑧 and the

curve 𝛾𝜁winds once around the straight line {𝜁0+𝑧𝑒3:𝑧∈ℝ},𝛾is homotopic to

the circle 𝐾(𝑅):={(𝑥−𝑥0)2+(𝑦−𝑦0)2=𝑅2,𝑧 =𝑧0},where𝜁0=𝑥0+𝑦0𝑒2+𝑧0𝑒3.

Then using equality (3.8), we have 𝐼2=2𝜋𝑖Φ(𝜁0).

Let us prove that 𝐼1=0.First,wechooseonthecurve𝛾two points 𝐴and

𝐵in which there are tangents to 𝛾, and we choose also two points 𝐴1,𝐵

1on the

circle 𝐾(𝜀) which is completely contained in the domain Ω. Let 𝛾1,𝛾2be connected

components of the set 𝛾∖{𝐴, 𝐵}.By𝐾1and 𝐾2we denote connected components

of the set 𝐾(𝜀)∖{𝐴1,𝐵

1}in such a way that after a choice of smooth arcs Γ1,Γ

each of the closed curves 𝛾1∪Γ2∪𝐾1∪Γ1and 𝛾2∪Γ1∪𝐾2∪Γ2will homotopic

to a point of the domain Ω ∖{(𝑥0,𝑦

0,𝑧):𝑧∈ℝ}.

Then it follows from Theorem 3.3 that



𝛾1

𝜁∪Γ2

𝜁∪𝐾1

𝜁∪Γ1

𝜁

(Φ(𝜁)−Φ(𝜁0)) (𝜁−𝜁0)−1𝑑𝜁 =0,(3.10)



𝛾2

𝜁∪Γ1

𝜁∪𝐾2

𝜁∪Γ2

𝜁

(Φ(𝜁)−Φ(𝜁0)) (𝜁−𝜁0)−1𝑑𝜁 =0.(3.11)

Inasmuch as each of the curves Γ1

𝜁,Γ

𝜁has diﬀerent orientations in the equal-

ities (3.10), (3.11), after addition of the mentioned equalities we obtain



𝛾𝜁

(Φ(𝜁)−Φ(𝜁0)) (𝜁−𝜁0)−1𝑑𝜁 =

𝐾𝜁(𝜀)

(Φ(𝜁)−Φ(𝜁0)) (𝜁−𝜁0)−1𝑑𝜁, (3.12)

where the curves 𝐾𝜁(𝜀), 𝛾𝜁have the same orientation.

The integrand in the right-hand side of the equality (3.12) is bounded by

a constant which does not depend on 𝜀. Therefore, passing to the limit in the

equality (3.12) as 𝜀→0, we obtain 𝐼1= 0 and the theorem is proved. □

196 S.A. Plaksa

Using formula (3.9), we obtain the Taylor expansion of a monogenic function

in the usual way (see, for example, [26, p. 107]). A uniqueness theorem for mono-

genic functions can also be proved in the same way as for holomorphic functions

of a complex variable (see, for example, [26, p. 110]).

3.5. Diﬀerent equivalent deﬁnitions of monogenic functions taking values

in the algebra 𝔸3

Thus, the following theorem giving diﬀerent equivalent deﬁnitions of monogenic

functions is true:

Theorem 3.6. AfunctionΦ:Ω

𝜁→𝔸3is monogenic in an arbitrary domain Ω𝜁if

and only if one of the following conditions is satisﬁed:

(I) the components 𝑈𝑘,𝑘= 1,3, of the decomposition (2.4) of the function Φare

diﬀerentiable in Ωand the conditions (2.5) are satisﬁed in the domain Ω𝜁;

(II) in every bal l ℧𝜁⊂Ω𝜁the function Φis expressed in the form (2.19),where

the triad of holomorphisms in the domain 𝑓(℧𝜁),functions𝐹,𝐹1,𝐹2,is

unique;

(III) the function Φis continuous in Ω𝜁and satisﬁes the equality (3.6) for every

triangle △𝜁such that △𝜁⊂Ω𝜁;

(IV) for every 𝜁0∈Ω𝜁there exists a neighborhood in which the function Φis

expressed as the sum of the power series (1.4),where𝔸=𝔸3.

4. Inﬁnite-dimensional commutative Banach algebras and spatial

potential ﬁelds

4.1. An inﬁnite-dimensional harmonic algebra

Note that it is impossible to obtain all solutions of the three-dimensional Laplace

equation (1.3) in the form of components of monogenic functions taking values

in commutative algebras of third rank. In particular, for every mentioned algebra

there exist spherical functions which are not components of speciﬁed hypercomplex

monogenic functions.

Indeed, it is well known that there exists (2𝑛+ 1) linearly independent ho-

mogeneous polynomials (of real variables 𝑥, 𝑦, 𝑧) of the degree 𝑛, which satisfy

(1.3). At the same time, in every algebra 𝔸over the ﬁeld ℂwith harmonic basis

{𝑒1,𝑒

2,𝑒

3}, the mentioned polynomials are real components of decomposition on

the vectors 𝑒1,𝑒

2,𝑒

3,𝑖𝑒

1,𝑖𝑒

2,𝑖𝑒

3of functions of the form 𝑎(𝑥𝑒1+𝑦𝑒2+𝑧𝑒3)𝑛,where

𝑎∈𝔸, but there are only six linearly independent real components for all 𝑛≥3.

Hence, for every harmonic algebra 𝔸of the third rank there exist spherical

functions which are not components of monogenic functions taking values in the

algebra 𝔸.

In the papers [4, 27, 28] we considered an inﬁnite-dimensional commutative

Banach algebra 𝔽over the ﬁeld of real numbers and established that any spherical

function is a component of some monogenic function taking values in this algebra.

Commutative Algebras and Equations of Mathematical Physics 197

We described relations between monogenic functions and harmonic vectors in the

space.

Consider an inﬁnite-dimensional commutative associative Banach algebra

𝔽:= {𝑔=∞



𝑘=1

𝑐𝑘𝑒𝑘:𝑐𝑘∈ℝ,∞



𝑘=1 ∣𝑐𝑘∣<∞}

over the ﬁeld ℝwith the norm ∥𝑔∥𝔽:= ∞



𝑘=1 ∣𝑐𝑘∣and the basis {𝑒𝑘}∞

𝑘=1,wherethe

multiplication table for elements of its basis is of the following form:

𝑒𝑛𝑒1=𝑒𝑛,𝑒

2𝑛+1𝑒2𝑛=1

2𝑒4𝑛∀𝑛≥1,

𝑒2𝑛+1𝑒2𝑚=1

2𝑒2𝑛+2𝑚−(−1)𝑚𝑒2𝑛−2𝑚∀𝑛>𝑚≥1,

𝑒2𝑛+1𝑒2𝑚=1

2𝑒2𝑛+2𝑚+(−1)𝑛𝑒2𝑚−2𝑛∀𝑚>𝑛≥1,

𝑒2𝑛+1𝑒2𝑚+1 =1

2𝑒2𝑛+2𝑚+1 +(−1)𝑚𝑒2𝑛−2𝑚+1∀𝑛≥𝑚≥1,

𝑒2𝑛𝑒2𝑚=1

2−𝑒2𝑛+2𝑚+1 +(−1)𝑚𝑒2𝑛−2𝑚+1∀𝑛≥𝑚≥1.

It is evident that here 𝑒1,𝑒

2,𝑒

3form a harmonic triad of vectors.

Let Ω be a domain in ℝ3and 𝜁=𝑥+𝑦𝑒2+𝑧𝑒3,where(𝑥, 𝑦, 𝑧 )∈Ω. Consider

the decomposition

Φ(𝜁)= ∞



𝑘=1

𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑒𝑘,(4.1)

of a function Φ : Ω𝜁→𝔽with respect to the basis {𝑒𝑘}∞

𝑘=1, where the functions

𝑈𝑘:Ω→ℝare diﬀerentiable in Ω.

In the following theorem we establish necessary and suﬃcient conditions for

a function Φ : Ω𝜁→𝔽to be monogenic in a domain Ω𝜁⊂𝐸3.

Theorem 4.1. Let a fu nction Φ:Ω

𝜁→𝔽be continuous in a domain Ω𝜁and the

functions 𝑈𝑘:Ω→ℝfrom the decomposition (4.1) be diﬀerentiab le in Ω.Inorder

that the function Φbe monogenic in the domain Ω𝜁, it is necessary and suﬃcient

that the conditions (2.5) be satisﬁed and the following relations be fulﬁlled in Ω:

∞



𝑘=1 

∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑥 <∞,(4.2)

lim

𝜀→0+0

∞



𝑘=1 𝑈𝑘(𝑥+𝜀ℎ1,𝑦+𝜀ℎ2,𝑧 +𝜀ℎ3)−𝑈𝑘(𝑥, 𝑦, 𝑧)−∂𝑈𝑘(𝑥, 𝑦, 𝑧 )

∂𝑥 𝜀ℎ1

−∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑦 𝜀ℎ2−∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑧 𝜀ℎ3𝜀−1=0 ∀ℎ1,ℎ

2,ℎ

3∈ℝ.(4.3)

198 S.A. Plaksa

Proof. Necessity. If the function (4.1) is monogenic in the domain Ω𝜁,thenfor

ℎ=𝑒1the equality (1.7) turns into the equality

Φ′(𝜁)=∂Φ(𝜁)

∂𝑥 ≡∞



𝑘=1

∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑥 𝑒𝑘,

and the relation (4.2) is fulﬁlled. Now, setting in series ℎ=𝑒2and ℎ=𝑒3in the

equality (1.7), we obtain the conditions (2.5).

Let us write the conditions (2.5) in expanded form:

∂𝑈1(𝑥, 𝑦, 𝑧)

∂𝑦 =−1

∂𝑈2(𝑥, 𝑦, 𝑧)

∂𝑥 ,

∂𝑈2(𝑥, 𝑦, 𝑧)

∂𝑦 =∂𝑈1(𝑥, 𝑦, 𝑧)

∂𝑥 +1

∂𝑈5(𝑥, 𝑦, 𝑧)

∂𝑥 ,

∂𝑈3(𝑥, 𝑦, 𝑧)

∂𝑦 =−1

∂𝑈4(𝑥, 𝑦, 𝑧)

∂𝑥 ,

∂𝑈2𝑘(𝑥, 𝑦, 𝑧)

∂𝑦 =1

∂𝑈2𝑘−1(𝑥, 𝑦, 𝑧)

∂𝑥 +1

∂𝑈2𝑘+3(𝑥, 𝑦, 𝑧 )

∂𝑥 ,𝑘=2,3,... ,

∂𝑈2𝑘+1(𝑥, 𝑦, 𝑧)

∂𝑦 =−1

∂𝑈2𝑘−2(𝑥, 𝑦, 𝑧)

∂𝑥 −1

∂𝑈2𝑘+2(𝑥, 𝑦, 𝑧)

∂𝑥 ,𝑘=2,3,... ,

∂𝑈1(𝑥, 𝑦, 𝑧)

∂𝑧 =−1

∂𝑈3(𝑥, 𝑦, 𝑧)

∂𝑥 ,

∂𝑈2(𝑥, 𝑦, 𝑧)

∂𝑧 =−1

∂𝑈4(𝑥, 𝑦, 𝑧)

∂𝑥 ,

∂𝑈3(𝑥, 𝑦, 𝑧)

∂𝑧 =∂𝑈1(𝑥, 𝑦, 𝑧)

∂𝑥 −1

∂𝑈5(𝑥, 𝑦, 𝑧)

∂𝑥 ,

∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑧 =1

∂𝑈𝑘−2(𝑥, 𝑦, 𝑧)

∂𝑥 −1

∂𝑈𝑘+2(𝑥, 𝑦, 𝑧)

∂𝑥 ,𝑘=4,5,... .

(4.4)

At last, for ℎ1,ℎ

2,ℎ

3∈ℝand 𝜀>0, writing ℎ:= ℎ1𝑒1+ℎ2𝑒2+ℎ3𝑒3and

taking into account the equalities (4.4), we have

Φ(𝜁+𝜀ℎ)−Φ(𝜁)𝜀−1−ℎΦ′(𝜁)

=∞



𝑘=1𝑈𝑘(𝑥+𝜀ℎ1,𝑦+𝜀ℎ2,𝑧+𝜀ℎ3)−𝑈𝑘(𝑥, 𝑦, 𝑧 )𝑒𝑘

−𝜀(ℎ1𝑒1+ℎ2𝑒2+ℎ3𝑒3)∞



𝑘=1

∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑥 𝑒𝑘𝜀−1

=𝜀−1∞



𝑘=1 𝑈𝑘(𝑥+𝜀ℎ1,𝑦+𝜀ℎ2,𝑧+𝜀ℎ3)−𝑈𝑘(𝑥, 𝑦, 𝑧 )

−∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑥 𝜀ℎ1−∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑦 𝜀ℎ2−∂𝑈𝑘(𝑥, 𝑦, 𝑧)

∂𝑧 𝜀ℎ3𝑒𝑘.

(4.5)

Commutative Algebras and Equations of Mathematical Physics 199

Therefore, inasmuch as the function (4.1) is monogenic in the domain Ω𝜁,the

relation (4.3) is fulﬁlled in Ω.

Suﬃciency. Let 𝜀>0andℎ:= ℎ1𝑒1+ℎ2𝑒2+ℎ3𝑒3,whereℎ1,ℎ

2,ℎ

3∈ℝ.Then

under the conditions (4.4) and (4.2) the equality (4.5) is true. Now, it follows from

the equality (4.5) and the relation (4.3) that the function (4.1) has the Gateaux

derivative Φ′(𝜁) for all 𝜁∈𝑄𝜁. The theorem is proved. □

Note that the conditions (4.4) are similar by nature to the Cauchy–Riemann

conditions for monogenic functions of complex variables, and the relations (4.2),

(4.3) are conditioned by the inﬁnite dimensionality of the algebra 𝔽.

It is clear that if the Gateaux derivative Φ′of monogenic function Φ : Ω𝜁→𝔽,

in turn, is a monogenic function in the domain Ω𝜁, then all components 𝑈𝑘of

decomposition (4.1) satisfy equation (1.3) in Ω in consequence of condition (1.5).

At the same time, the following statement is true even independently of the relation

between solutions of the system of equations (4.4) and monogenic functions.

Theorem 4.2. If the functions 𝑈𝑘:Ω→ℝhave continuous second-order par-

tial derivatives in a domain Ωand satisfy the conditions (4.4), then they satisfy

equat ion (1.3) in 𝑄.

To prove Theorem 4.2 it is easy to show that if the functions 𝑈𝑘are doubly

continuously diﬀerentiable in the domain Ω, then the equalities Δ𝑈𝑘(𝑥, 𝑦, 𝑧 )=0

for 𝑘=1,2,... are corollaries of the system (4.4).

Note that the algebra 𝔽is isomorphic to the algebra Fof absolutely conver-

gent trigonometric Fourier series

𝑔(𝜏)=𝑎0+∞



𝑘=1𝑎𝑘𝑖𝑘cos 𝑘𝜏 +𝑏𝑘𝑖𝑘sin 𝑘𝜏

with real coeﬃcients 𝑎0,𝑎

𝑘,𝑏

𝑘and the norm ∥𝑔∥F:= ∣𝑎0∣+∞



𝑘=1∣𝑎𝑘∣+∣𝑏𝑘∣.In

this case, we have the isomorphism 𝑒2𝑘−1←→ 𝑖𝑘−1cos (𝑘−1)𝜏,𝑒2𝑘←→ 𝑖𝑘sin 𝑘𝜏

between basic elements.

Let us write the expansion of a power function of the variable 𝜉=𝑥𝑒1+

𝑦𝑒2+𝑧𝑒3in the basis {𝑒𝑘}∞

𝑘=1, using spherical coordinates 𝜌, 𝜃, 𝜙 which have the

following relations with 𝑥, 𝑦, 𝑧 :

𝑥=𝜌cos 𝜃, 𝑦 =𝜌sin 𝜃sin 𝜙, 𝑧 =𝜌sin 𝜃cos 𝜙. (4.6)

In view of the isomorphism of the algebras 𝔽and F, the construction of expansions

of this sort is reduced to the determination of relevant Fourier coeﬃcients. So, we

have

𝜁𝑛=𝜌𝑛𝑃𝑛(cos 𝜃)𝑒1+2

𝑛



𝑚=1

𝑛!

(𝑛+𝑚)!𝑃𝑚

𝑛(cos 𝜃)sin 𝑚𝜙 𝑒2𝑚+cos𝑚𝜙 𝑒2𝑚+1,

(4.7)

200 S.A. Plaksa

where 𝑛is a positive integer, 𝑃𝑛and 𝑃𝑚

𝑛are Legendre polynomials and associated

Legendre polynomials, respectively, namely:

𝑃𝑛(𝑡):= 1

2𝑛𝑛!

𝑑𝑛

𝑑𝑡𝑛(𝑡2−1)𝑛,𝑃

𝑚

𝑛(𝑡):=(1−𝑡2)𝑚/2𝑑𝑚

𝑑𝑡𝑚𝑃𝑛(𝑡).(4.8)

We obtain in exactly the same way the following expansion of the exponential

function:

𝑒𝜁=𝑒𝜌cos 𝜃𝐽0(𝜌sin 𝜃)𝑒1+2∞



𝑚=1

𝐽𝑚(𝜌sin 𝜃)sin 𝑚𝜙 𝑒2𝑚+cos𝑚𝜙 𝑒2𝑚+1,

where 𝐽𝑚are Bessel functions, namely:

𝐽𝑚(𝑡):= (−1)𝑚

𝜋



𝑒𝑖𝑡 cos 𝜏cos 𝑚𝜏 𝑑𝜏 . (4.9)

Thus, 2𝑛+ 1 linearly independent spherical functions of the 𝑛th power are

components of the expansion (4.7) of the function 𝜁𝑛. Using the expansion (4.7)

and rules of multiplication for basic elements of the algebra 𝔽,itiseasytoprove

the following statement.

Theorem 4.3. Every spherical function

𝜌𝑛𝑎𝑛,0𝑃𝑛(cos 𝜃)+

𝑛



𝑚=1𝑎𝑛,𝑚 cos 𝑚𝜙 +𝑏𝑛,𝑚 sin 𝑚𝜙𝑃𝑚

𝑛(cos 𝜃)

where 𝑎𝑛,0,𝑎

𝑛,𝑚,𝑏

𝑛,𝑚 ∈ℝ, is the ﬁrst component of expansion of the monogenic

function 𝑎𝑛,0𝑒1+

𝑛



𝑚=1

(−1)𝑚(𝑛+𝑚)!

𝑛!𝑏𝑛,𝑚 𝑒2𝑚+𝑎𝑛,𝑚 𝑒2𝑚+1𝜉𝑛

in the basis {𝑒𝑘}∞

𝑘=1,where𝜉:= 𝑥𝑒1+𝑦𝑒2+𝑧𝑒3,and 𝑥, 𝑦, 𝑧 have the relations

(4.6) with the spherical coordinates 𝜌, 𝜃, 𝜙.

4.2. Relation between the system (4.4) and harmonic vectors

Let us specify a relation between solutions of the system (4.4) and spatial potential

ﬁelds.

Theorem 4.4. Every solution of the system (4.4) in a domain Ω⊂ℝ3generates a

harmonic vector V:= (𝑈1,−1

2𝑈2,−1

2𝑈3)in Ω.

Proof. The equalities (1.2) for a vector V:= (𝑈1,−1

2𝑈2,−1

2𝑈3) are corollaries

of the system (4.4). Really, the system (4.4) contains the third equation and the

fourth equation from (1.2). The second equation from (1.2) is a corollary of the

third condition and the seventh condition of the system (4.4). Finally, the ﬁrst

equation from (1.2) is a corollary of the second condition and the eighth condition

of the system (4.4). The theorem is proved. □

Commutative Algebras and Equations of Mathematical Physics 201

Theorem 4.5. For any function 𝑈1:Ω→ℝharmonic in a simply connected

domain Ω⊂ℝ3there exist harmonic functions 𝑈𝑘:Ω→ℝ,𝑘=2,3,...,such

that the conditions (4.4) are fulﬁlled in Ω.

Proof. Let us add and subtract the second condition and the eighth condition as

well as the third condition and the seventh condition of the system (4.4). Let us

also add and subtract the fourth condition for 𝑘=𝑚and the ninth condition of

the system (4.4) for 𝑘=2𝑚+1, where 𝑚=2,3,... . Let us else add and subtract

the ﬁfth condition for 𝑘=𝑚and the ninth condition of system (4.4) for 𝑘=2𝑚,

where 𝑚=2,3,... . Thus, we rewrite the system (4.4) in the following equivalent

form:

∂𝑈1

∂𝑥 −1

∂𝑈2

∂𝑦 −1

∂𝑈3

∂𝑧 =0,

∂𝑈3

∂𝑦 −∂𝑈2

∂𝑧 =0,

∂𝑈1

∂𝑦 +1

∂𝑈2

∂𝑥 =0,

∂𝑈1

∂𝑧 +1

∂𝑈3

∂𝑥 =0,

∂𝑈2𝑘

∂𝑥 =−∂𝑈2𝑘−2

∂𝑧 −∂𝑈2𝑘−1

∂𝑦 ,

∂𝑈2𝑘+1

∂𝑥 =∂𝑈2𝑘−2

∂𝑦 −∂𝑈2𝑘−1

∂𝑧 ,

∂𝑈2𝑘

∂𝑧 −∂𝑈2𝑘+1

∂𝑦 =∂𝑈2𝑘−2

∂𝑥 ,

∂𝑈2𝑘

∂𝑦 +∂𝑈2𝑘+1

∂𝑧 =∂𝑈2𝑘−1

∂𝑥 𝑘=2,3,... .

(4.10)

First of all, note that there exists a harmonic vector V0:= (𝑈1,𝑣

2,𝑣

3)in

the domain Ω. Moreover, for any vector V:= (𝑈1,𝑣

2,𝑣

3) harmonic in Ω, the

components 𝑣2,𝑣

3are determined accurate within the real part and the imaginary

part of any function 𝑓1(𝑡) holomorphic in the domain {𝑡=𝑧+𝑖𝑦 :(𝑥, 𝑦, 𝑧)∈Ω}

of the complex plane, i.e., the equalities

𝑣2(𝑥, 𝑦, 𝑧)=𝑣0

2(𝑥, 𝑦, 𝑧 )+Re𝑓1(𝑧+𝑖𝑦),𝑣

3(𝑥, 𝑦, 𝑧 )=𝑣0

3(𝑥, 𝑦, 𝑧 )+Im𝑓1(𝑧+𝑖𝑦)

are true for all (𝑥, 𝑦, 𝑧)∈Ω. Then, using Theorem 4.4, we ﬁnd the functions 𝑈2

and 𝑈3,namely:

𝑈2:= −2𝑣2,𝑈

3:= −2𝑣3.

Now, let us show that the last four conditions of the system (4.10) allow us to

determine the functions 𝑈2𝑘,𝑈2𝑘+1, if the functions 𝑈2,𝑈

3,...,𝑈

2𝑘−1are already

determined. Really, integrating the ﬁfth equation and the sixth equation of the

202 S.A. Plaksa

system (4.10), we obtain the expressions

𝑈2𝑘(𝑥, 𝑦, 𝑧 )=−

𝑥



𝑥0∂𝑈2𝑘−2(𝜏,𝑦, 𝑧)

∂𝑧 +∂𝑈2𝑘−1(𝜏,𝑦, 𝑧)

∂𝑦 𝑑𝜏 +𝑢2𝑘(𝑧, 𝑦),

𝑈2𝑘+1(𝑥, 𝑦, 𝑧)=

𝑥



𝑥0∂𝑈2𝑘−2(𝜏,𝑦, 𝑧)

∂𝑦 −∂𝑈2𝑘−1(𝜏,𝑦, 𝑧)

∂𝑧 𝑑𝜏 +𝑢2𝑘+1 (𝑧, 𝑦)

for (𝑥, 𝑦, 𝑧 ) belonging to a certain neighborhood 𝒩⊂Ωofanypoint(𝑥0,𝑦

0,𝑧

0)∈Ω.

Substituting these expressions into the seventh equation and eighth equation

of the system (4.10) and taking into account that 𝑈2𝑘−2,𝑈2𝑘−1are harmonic func-

tions in the domain 𝒩, we obtain the following inhomogeneous Cauchy–Riemann

system (see, for example, [29]) for ﬁnding the functions 𝑢2𝑘,𝑢2𝑘+1:

∂𝑢2𝑘(𝑧, 𝑦)

∂𝑧 −∂𝑢2𝑘+1 (𝑧, 𝑦)

∂𝑦 =∂𝑈2𝑘−2(𝑥, 𝑦, 𝑧)

∂𝑥 𝑥=𝑥0

∂𝑢2𝑘(𝑧, 𝑦)

∂𝑦 +∂𝑢2𝑘+1 (𝑧, 𝑦)

∂𝑧 =∂𝑈2𝑘−1(𝑥, 𝑦, 𝑧)

∂𝑥 𝑥=𝑥0

(4.11)

Solutions of the system (4.11) are determined accurate within the real part and the

imaginary part of any function holomorphic in the domain {𝑡=𝑧+𝑖𝑦 :(𝑥, 𝑦, 𝑧 )∈

𝒩} of the complex plane. Therefore, inasmuch as the domain Ω is simply con-

nected, taking into account the uniqueness theorem for spatial harmonic functions,

it is easy to continue the functions 𝑈2𝑘,𝑈2𝑘+1 deﬁned in the neighborhood 𝒩into

the domain Ω. The theorem is proved. □

Let us note that the functions 𝑈2𝑚,𝑈2𝑚+1 satisfying the last four conditions

of the system (4.10) for 𝑘=𝑚≥2 are determined accurate within the real

part and the imaginary part of any function 𝑓𝑚(𝑡) holomorphic in the domain

{𝑡=𝑧+𝑖𝑦 :(𝑥, 𝑦 , 𝑧)∈Ω}of the complex plane, i.e., the equalities

𝑈2𝑚(𝑥, 𝑦, 𝑧 )=𝑈0

2𝑚(𝑥, 𝑦, 𝑧 )+Re𝑓𝑚(𝑧+𝑖𝑦),

𝑈2𝑚+1(𝑥, 𝑦, 𝑧)=𝑈0

2𝑚+1(𝑥, 𝑦, 𝑧)+Im𝑓𝑚(𝑧+𝑖𝑦)

are true for all (𝑥, 𝑦, 𝑧)∈Ω, where 𝑈0

2𝑚,𝑈0

2𝑚+1 are functions forming together

with the functions 𝑈1,𝑈

2,...,𝑈

2𝑚−1a particular solution of the system (4.10), in

which 𝑘=2,3,...,𝑚.

4.3. Monogenic functions and axial-symmetric potential ﬁelds

In the case where a spatial potential ﬁeld is symmetric with respect to the axis

𝑂𝑥, a potential function 𝑢(𝑥, 𝑦, 𝑧 ) satisfying equation (1.3) is also symmetric with

respect to the axis 𝑂𝑥, i.e., 𝑢(𝑥, 𝑦, 𝑧)=𝜑(𝑥, 𝑟)=𝜑(𝑥, −𝑟), where 𝑟:= 𝑦2+𝑧2,

and 𝜑is known as the axial-symmetric potential. Then in a meridian plane 𝑥𝑂𝑟

there exists a function 𝜓(𝑥, 𝑟)knownastheStokes ﬂow function such that the

Commutative Algebras and Equations of Mathematical Physics 203

functions 𝜑and 𝜓satisfy the following system of equations degenerating on the

axis 𝑂𝑥:

𝑟∂𝜑(𝑥, 𝑟)

∂𝑥 =∂𝜓(𝑥, 𝑟)

∂𝑟 ,𝑟

∂𝜑(𝑥, 𝑟)

∂𝑟 =−∂𝜓(𝑥, 𝑟)

∂𝑥 .(4.12)

Under the condition that there exist continuous second-order partial deriva-

tives of the functions 𝜑(𝑥, 𝑟)and𝜓(𝑥, 𝑟), the system (4.12) implies the equation

𝑟Δ𝜑(𝑥, 𝑟)+∂𝜑(𝑥, 𝑟)

∂𝑟 = 0 (4.13)

for the axial-symmetric potential and the equation

𝑟Δ𝜓(𝑥, 𝑟)−∂𝜓(𝑥, 𝑟)

∂𝑟 = 0 (4.14)

for the Stokes ﬂow function, where Δ := ∂2

∂𝑥2+∂2

∂𝑟2.

We proved in the papers [4, 30] that in a domain convex in the direction of

the axis 𝑂𝑟 the functions 𝜑and 𝜓can be constructed by means of components of

principal extensions of analytic functions of a complex variable into a correspond-

ing domain of a special two-dimensional vector manifold in an inﬁnite-dimensional

commutative Banach algebra ℍℂover the ﬁeld of complex numbers.

In such a way for solutions of the system (4.12) we obtained integral ex-

pressions which were generalized for domains of general form (see [4, 31]). Using

integral expressions for solutions of the system (4.12), in the papers [4, 32–35] we

developed methods for solving boundary problems for axial-symmetric potentials

and Stokes ﬂow functions that have various applications in mathematical physics.

In particular, the developed methods are applicable for solving a boundary prob-

lem about a streamline of the ideal incompressible ﬂuid along an axial-symmetric

body (see [4, 36]).

Now, let us consider a subalgebra

ℍ:= {𝑎=∞



𝑘=1

𝑎𝑘𝑒2𝑘−1:𝑎𝑘∈ℝ,∞



𝑘=1 ∣𝑎𝑘∣<∞}

of the algebra 𝔽. In the paper [37] I. Mel’nichenko oﬀered the algebra ℍfor de-

scribing spatial axial-symmetric potential ﬁelds.

As in the papers [30, 38], consider a complexiﬁcation

ℍℂ:= ℍ⊕𝑖ℍ≡{𝑐=𝑎+𝑖𝑏 :𝑎,𝑏 ∈ℍ}

of the algebra ℍsuch that the norm of element 𝑔:= ∞



𝑘=1

𝑐𝑘𝑒2𝑘−1∈ℍℂis given by

means of the equality ∥𝑔∥ℍℂ:= ∞



𝑘=1 ∣𝑐𝑘∣.

204 S.A. Plaksa

The algebra ℍℂis isomorphic to the algebra Fcos of absolutely convergent

trigonometric Fourier series

𝑐(𝜏)= ∞



𝑘=1

𝑐𝑘𝑖𝑘−1cos(𝑘−1)𝜏

with complex coeﬃcients 𝑐𝑘and the norm ∥𝑐∥Fcos := ∞



𝑘=1 ∣𝑐𝑘∣. In this case, we have

the isomorphism 𝑒2𝑘−1←→ 𝑖𝑘−1cos(𝑘−1)𝜏between basic elements.

The structure of maximal ideals of the algebra of absolutely convergent com-

plex Fourier series is described in the monograph [39]. This description allows us

to make a conclusion about the structure of maximal ideals of the algebra ℍℂand

about the fact that there do not exist inverse elements in ℍℂfor the basic elements

𝑒𝑘,where𝑘=2,3,....

In particular, the set

ℐ0:=𝑔∈ℍℂ:∞



𝑘=1

(−1)𝑘(Re𝑐2𝑘−1−Im𝑐2𝑘)=0,∞



𝑘=1

(−1)𝑘(Re𝑐2𝑘+Im𝑐2𝑘−1)=0

is a maximum ideal of the algebra ℍℂ.

Let 𝑓ℐ0:ℍℂ→ℂbe the linear functional deﬁned by the equality

𝑓ℐ0∞



𝑘=1

𝑐𝑘𝑒2𝑘−1

:= −∞



𝑘=1

(−1)𝑘(Re 𝑐2𝑘−1−Im 𝑐2𝑘)−𝑖∞



𝑘=1

(−1)𝑘(Re 𝑐2𝑘+Im 𝑐2𝑘−1).

It is evident that the ideal ℐ0is the kernel of the functional 𝑓ℐ0and 𝑓(𝑒1)=1,

i.e., 𝑓ℐ0is a continuous multiplicative functional.

Consider the Cartesian plane 𝜇:= {𝜁=𝑥𝑒1+𝑟𝑒3:𝑥, 𝑟 ∈ℝ}. For a domain

𝐸⊂ℝ2we use coordinated denotations for congruent domains of the plane 𝜇and

the complex plane ℂ,namely:𝐸𝜁:= {𝜁=𝑥𝑒1+𝑟𝑒3:(𝑥, 𝑟)∈𝐸}⊂𝜇and

𝐸𝑧:= {𝑧=𝑥+𝑖𝑟 :(𝑥, 𝑟)∈𝐸}⊂ℂ.

We say that a continuous function Φ : 𝐸𝜁→ℍℂis monogenic in a domain

𝐸𝜁if Φ is diﬀerentiable in the sense of Gateaux in every point of 𝐸𝜁, i.e., for every

𝜁∈𝐸𝜁there exists an element Φ′(𝜁)∈ℍℂsuch that the equality (1.7) is fulﬁlled

for all ℎ∈𝜇.

The proof of the following theorem is similar to the proof of Theorem 4.1.

Theorem 4.6. Le t a fun ction Φ:𝐸𝜁→ℍℂbe continuous in a domain 𝐸𝜁⊂𝜇and

the functions 𝑈𝑘:𝐸→ℂfrom the decomposition

Φ(𝑥𝑒1+𝑟𝑒3)= ∞



𝑘=1

𝑈𝑘(𝑥, 𝑟)𝑒2𝑘−1

be diﬀeren tiable i n 𝐸. In order that the function Φbe monogenic in the domain

𝐸𝜁, it is necessary and suﬃcient that the conditions

∂Φ(𝜁)

∂𝑦 =∂Φ(𝜁)

∂𝑥 𝑒3(4.15)

Commutative Algebras and Equations of Mathematical Physics 205

be satisﬁed and that the following relations be fulﬁlled in 𝐸:

∞



𝑘=1 

∂𝑈𝑘(𝑥, 𝑟)

∂𝑥 <∞,(4.16)

lim

𝜀→0+0

∞



𝑘=1 𝑈𝑘(𝑥+𝜀ℎ1,𝑟+𝜀ℎ2,)−𝑈𝑘(𝑥, 𝑟)−∂𝑈𝑘(𝑥, 𝑟)

∂𝑥 𝜀ℎ1

−∂𝑈𝑘(𝑥, 𝑟)

∂𝑟 𝜀ℎ2𝜀−1=0 ∀ℎ1,ℎ

2∈ℝ.(4.17)

Note that the conditions (4.15) rewritten in expanded form

∂𝑈1(𝑥, 𝑟)

∂𝑟 =−1

∂𝑈2(𝑥, 𝑟)

∂𝑥 ,

∂𝑈2(𝑥, 𝑟)

∂𝑟 =∂𝑈1(𝑥, 𝑟)

∂𝑥 −1

∂𝑈3(𝑥, 𝑟)

∂𝑥 ,

∂𝑈𝑘(𝑥, 𝑟)

∂𝑟 =1

∂𝑈𝑘−1(𝑥, 𝑟)

∂𝑥 −1

∂𝑈𝑘+1(𝑥, 𝑟)

∂𝑥 ,𝑘=3,4,... ,

are similar by nature to the Cauchy–Riemann conditions for monogenic functions

of a complex variable, and the relations (4.16), (4.17) are conditioned by the inﬁnite

dimensionality of the algebra ℍℂ.

In the papers [4, 30] we established relations between monogenic functions

taking values in the algebra ℍℂand solutions of the system (4.12) in so-called

proper domains.

We ca l l 𝐸𝑧aproper domain in ℂ, provided that for every 𝑧∈𝐸𝑧with

Im 𝑧∕= 0 the domain 𝐸𝑧contains the segment connecting points 𝑧and ¯𝑧.Inthis

case 𝐸𝜁is also called a proper domain in 𝜇.

Let 𝐴be the linear operator which assigns the function 𝐹:𝐸𝑧→ℂto every

function Φ : 𝐸𝜁→ℍℂby the formula 𝐹(𝑧):=𝑓ℐ0(Φ(𝜁)), where 𝑧=𝑥+𝑖𝑟 and

𝜁=𝑥𝑒1+𝑟𝑒3. It is easy to prove that if Φ is a monogenic function in 𝐸𝜁,then𝐹

is an analytic function in 𝐸𝑧.

Theorem 4.7 ([4, 30]). If 𝐸𝜁is a proper domain in 𝜇, then every monogenic func-

tion Φ:𝐸𝜁→ℍℂcan be expressed in the form

Φ(𝜁)= 1

2𝜋𝑖 

𝛾

(𝐴Φ)(𝑡)(𝑡𝑒1−𝜁)−1𝑑𝑡 +Φ

0(𝜁)∀𝜁∈𝐸𝜁,(4.18)

where 𝛾is an arbitrary closed Jordan rectiﬁable curve in 𝐷𝑧that embraces the

segment connecting the points 𝑧=𝑓ℐ0(𝜁)and ¯𝑧,andΦ0:𝐸𝜁→ℐ

0is a monogenic

function taking values in the ideal ℐ0.

Note that under the conditions of Theorem 4.7 the integral in the equality

(4.18) is the principal extension (see [8, p. 165]) of an analytic function 𝐹=

𝐴Φ into the domain 𝐷𝜁. Thus, the algebra of monogenic functions in 𝐸𝜁can be

206 S.A. Plaksa

decomposed into the direct sum of the algebra of principal extensions of complex-

valued analytic functions and the algebra of monogenic functions taking values in

the ideal ℐ0.

In the papers [4, 30], for every function 𝐹:𝐸𝑧→ℂanalytic in a proper

domain 𝐸𝑧we constructed explicitly the decomposition of the principal extension

of 𝐹into the domain 𝐸𝜁with respect to the basis {𝑒2𝑘−1}∞

𝑘=1:

2𝜋𝑖 

𝛾

(𝑡𝑒1−𝜁)−1𝐹(𝑡)𝑑𝑡 =𝑈1(𝑥, 𝑟)𝑒1+2 ∞



𝑘=2

𝑈𝑘(𝑥, 𝑟)𝑒2𝑘−1,(4.19)

where

𝑈𝑘(𝑥, 𝑟):= 1

2𝜋𝑖 

𝛾

𝐹(𝑡)

(𝑡−𝑧)(𝑡−¯𝑧)(𝑡−𝑧)(𝑡−¯𝑧)−(𝑡−𝑥)

𝑟𝑘−1

𝑑𝑡 , (4.20)

𝜁=𝑥𝑒1+𝑟𝑒3and 𝑧=𝑥+𝑖𝑟 for (𝑥, 𝑟)∈𝐸, and the curve 𝛾has the same

properties as in Theorem 4.7, and (𝑡−𝑧)(𝑡−¯𝑧) is a continuous branch of this

function analytic with respect to 𝑡outside of the segment mentioned in Theorem

4.7. Note that for every 𝑧∈𝐸𝑧with Im 𝑧= 0, we deﬁne (𝑡−𝑧)(𝑡−¯𝑧):=𝑡−𝑧.

In the following theorem we describe relations between principal extensions

of analytic functions into the plane 𝜇and solutions of the system (4.12).

Theorem 4.8 ([4, 30]). If 𝐹:𝐸𝑧→ℂis an analytic function in a proper domain

𝐸𝑧, then the ﬁrst and the second components of principal extension (4.19) of func-

tion 𝐹into the domain 𝐸𝜁generate the solutions 𝜑and 𝜓of system (4.12) in 𝐸

by the formulas

𝜑(𝑥, 𝑟)=𝑈1(𝑥, 𝑟),𝜓(𝑥, 𝑟)=𝑟𝑈

2(𝑥, 𝑟).(4.21)

From the relations (4.19)–(4.21) it follows that the functions

𝜑(𝑥, 𝑟)= 1

2𝜋𝑖 

𝛾

𝐹(𝑡)

(𝑡−𝑧)(𝑡−¯𝑧)𝑑𝑡 , (4.22)

𝜓(𝑥, 𝑟)=−1

2𝜋𝑖 

𝛾

𝐹(𝑡)(𝑡−𝑥)

(𝑡−𝑧)(𝑡−¯𝑧)𝑑𝑡 , 𝑧 =𝑥+𝑖𝑟 , (4.23)

are solutions of system (4.12) in the domain 𝐸.

In the following theorem we describe relations between components 𝑈𝑘of

hypercomplex analytic function (4.19) and solutions of elliptic equations degener-

ating on the axis 𝑂𝑥.

Theorem 4.9 ([40, 41]). If 𝐹:𝐸𝑧→ℂis an analytic function in a proper domain

𝐸𝑧, then the components 𝑈𝑘of principal extension (4.19) of function 𝐹into the

domain 𝐸𝜁satisfy the equations

𝑟2Δ𝑈𝑘(𝑥, 𝑟)+𝑟∂𝑈𝑘(𝑥, 𝑟)

∂𝑟 −(𝑘−1)2𝑈𝑘(𝑥, 𝑟)=0,𝑘=1,2,... ,

Commutative Algebras and Equations of Mathematical Physics 207

in the domain 𝐸. In addition, the function

𝜓𝑘(𝑥, 𝑟):=𝑟𝑘−1𝑈𝑘(𝑥, 𝑟)

is a solution in 𝐸of the equation

𝑟Δ𝜓𝑘(𝑥, 𝑟)−(2𝑘−3) ∂𝜓𝑘(𝑥, 𝑟)

∂𝑟 =0,𝑘=1,2,... .

In the following theorem we establish an expression of generalized axial-

symmetric potential via components 𝑈𝑘of the hypercomplex analytic function

(4.19).

Theorem 4.10 ([42]). If 𝐹:𝐸𝑧→ℂis an analytic function in a proper domain

𝐸𝑧, then the function

𝑢(𝑥, 𝑟):= 1+ ∞



𝑛=1

(𝑚−1)(𝑚−3) ...(𝑚−4𝑛+1)

24𝑛(2𝑛!) 𝐶𝑛

2𝑛𝑈1(𝑥, 𝑟) (4.24)

+∞



𝑘=1 ∞



𝑛=𝑘

(𝑚−1)(𝑚−3) ...(𝑚−4𝑛+1)

24𝑛(2𝑛!) 𝐶𝑛−𝑘

2𝑛𝑈4𝑘+1(𝑥, 𝑟)

+∞



𝑘=0 ∞



𝑛=𝑘

(𝑚−1)(𝑚−3) ...(𝑚−4𝑛−1)

24𝑛+2(2𝑛+1)! 𝐶𝑛−𝑘

2𝑛+1𝑈4𝑘+3 (𝑥, 𝑟),

where 𝐶𝑘

𝑛are the binomial coeﬃcients, satisﬁes the equation

Δ𝑢(𝑥, 𝑟)+𝑚

𝑟

∂𝑢(𝑥, 𝑟)

∂𝑟 = 0 (4.25)

on the set {(𝑥, 𝑟)∈𝐷:𝑟∕=0}for 𝑚≥1.

Furthermore, the function (4.24) is expressed in the form

𝑢(𝑥, 𝑟)= 2(𝑚−3)/2

𝜋𝑖 ∣𝑟∣𝑚−1

𝛾′

𝐹(𝑡)

(𝑡−𝑧)(𝑡−¯𝑧)[(𝑡−𝑧)(𝑡−¯𝑧)](𝑚−1)/2𝑑𝑡 ,

where 𝛾′is an arbitrary closed Jordan rectiﬁable curve in 𝐸𝑧that contains the

point s 𝑧and ¯𝑧and embraces the set {𝑥+𝑖𝜂 :∣𝜂∣<∣𝑟∣},and[(𝑡−𝑧)(𝑡−¯𝑧)](𝑚−1)/2

is a continuous branch of this function, analytic with respect to 𝑡outside of the cut

{𝑥+𝑖𝜂 :∣𝜂∣≥∣𝑟∣}.

Let us write expansions with respect to the basis {𝑒2𝑘−1}∞

𝑘=1 of some ele-

mentary analytic functions of the variable 𝜁=𝑥𝑒1+𝑟𝑒3(note that in view of the

isomorphism between algebras ℍℂand Fcos, the construction of expansions of this

sort is reduced to a determination of relevant Fourier coeﬃcients). The expansion

of a power function has the form

𝜁𝑛=(𝑥2+𝑟2)𝑛/2𝑃𝑛(cos 𝜗)𝑒1+2

𝑛



𝑘=1

(sgn 𝑟)𝑘𝑛!

(𝑛+𝑘)! 𝑃𝑘

𝑛(cos 𝜗)𝑒2𝑘+1,

208 S.A. Plaksa

where 𝑛is a positive integer, cos 𝜗:= 𝑥(𝑥2+𝑟2)−1/2,

sgn 𝑟:= 1,for 𝑟≥0,

−1,for 𝑟<0,

and Legendre polynomials 𝑃𝑛and associated Legendre polynomials 𝑃𝑚

𝑛are deﬁned

to be the equalities (4.8).

For the functions 𝑒𝜁,sin𝜁and cos 𝜁we have

𝑒𝜁=𝑒𝑥𝐽0(𝑟)𝑒1+2 ∞



𝑘=1

𝐽𝑘(𝑟)𝑒2𝑘+1,

sin 𝜁=sin𝑥𝐽0(𝑖𝑟)𝑒1+2 ∞



𝑘=1

𝐽2𝑘(𝑖𝑟)𝑒4𝑘+1−2𝑖cos 𝑥∞



𝑘=1

𝐽2𝑘−1(𝑖𝑟)𝑒4𝑘−1,

cos 𝜁=cos𝑥𝐽0(𝑖𝑟)𝑒1+2 ∞



𝑘=1

𝐽2𝑘(𝑖𝑟)𝑒4𝑘+1+2𝑖sin 𝑥∞



𝑘=1

𝐽2𝑘−1(𝑖𝑟)𝑒4𝑘−1,

where Bessel functions 𝐽𝑚are deﬁned by the equality (4.9).

For the functions 𝜁−1and Ln 𝜁we obtain

𝜁−1=









√𝑥2+𝑟2𝑒1+2 ∞



𝑘=1

(−1)𝑘√𝑥2+𝑟2−𝑥

𝑟𝑘

𝑒2𝑘+1for 𝑥>0,

−1

√𝑥2+𝑟2𝑒1+2 ∞



𝑘=1√𝑥2+𝑟2+𝑥

𝑟𝑘

𝑒2𝑘+1for 𝑥<0,

Ln 𝜁=











ln √𝑥2+𝑟2+𝑥

2+2𝑚𝜋𝑖𝑒1

+2 ∞



𝑘=1

(−1)𝑘+1

𝑘√𝑥2+𝑟2−𝑥

𝑟𝑘

𝑒2𝑘+1 for 𝑥>0,

ln √𝑥2+𝑟2−𝑥

2+(2𝑚+1)𝜋𝑖𝑒1

+2 ∞



𝑘=1

𝑘√𝑥2+𝑟2+𝑥

𝑟𝑘

𝑒2𝑘+1 for 𝑥<0,

where 𝑚is an integer number. In this case, the functions 𝜁−1and Ln 𝜁are not

deﬁned for 𝑥=0.

4.4. Integral expressions for axial-symmetric potential and

the Stokes ﬂow function

In the papers [4, 31] we generalized integral expressions (4.22) and (4.23) for the

axial-symmetric potential and the Stokes ﬂow function, respectively, to the case

of an arbitrary simply connected domain symmetric with respect to the axis 𝑂𝑥.

Below in Section 4, 𝐸is a bounded simply connected domain symmetric with

respect to the axis 𝑂𝑥. For every 𝑧∈𝐸𝑧with Im 𝑧∕= 0, we ﬁx an arbitrary Jordan

rectiﬁable curve Γ𝑧¯𝑧in 𝐸𝑧which connects the points 𝑧and ¯𝑧. In this case, let

Commutative Algebras and Equations of Mathematical Physics 209

(𝑡−𝑧)(𝑡−¯𝑧) be a continuous branch of this function analytic with respect to 𝑡

outside of the cut along Γ𝑧¯𝑧. As earlier, for every 𝑧∈𝐸𝑧with Im 𝑧= 0 we deﬁne

(𝑡−𝑧)(𝑡−¯𝑧):=𝑡−𝑧.

Theorem 4.11 ([31, 35]). If 𝐹is an analytic function in the domain 𝐸𝑧,then

the functions (4.22) and (4.23) are solutions of system (4.12) in 𝐸,where𝛾is

an arbitrary closed Jordan rectiﬁable curve in 𝐸𝑧which embraces Γ𝑧¯𝑧.Functions

(4.22) and (4.23) are also solutions of equations (4.13) and (4.14), respectively.

It is evident that if the boundary ∂𝐸𝑧is a Jordan rectiﬁable curve and the

function 𝐹belongs to the Smirnov class 𝐸1(see [43], p. 205) in the domain 𝐸𝑧,

then the formulas (4.22) and (4.23) can be transformed to the form

𝜑(𝑥, 𝑟)= 1

2𝜋𝑖 

∂𝐸𝑧

𝐹(𝑡)

(𝑡−𝑧)(𝑡−¯𝑧)𝑑𝑡 , (4.26)

𝜓(𝑥, 𝑟)=−1

2𝜋𝑖 

∂𝐸𝑧

𝐹(𝑡)(𝑡−𝑥)

(𝑡−𝑧)(𝑡−¯𝑧)𝑑𝑡 , 𝑧 =𝑥+𝑖𝑟 , (4.27)

for all (𝑥, 𝑟)∈𝐸,where𝐹(𝑡) is the angular boundary value of the function 𝐹

which is known to exist almost everywhere on ∂𝐸𝑧.

If 𝑧∈𝐸𝑧with Im 𝑧∕=0and𝛼∈ℝ,then(𝑡−𝑧)(𝑡−¯𝑧)𝛼is understood as a

continuous branch of the function 𝐿(𝑡):=(𝑡−𝑧)(𝑡−¯𝑧)𝛼analytic with respect

to 𝑡outside of the cut along a Jordan curve that successively connects the points

𝑧,∞and ¯𝑧, and the only common points of which with the set Γ𝑧¯𝑧∪ℝare the

points 𝑧and ¯𝑧.Inthiscase𝐿(𝑡)>0 for all 𝑡> max

𝜏∈Γ𝑧¯𝑧

Re 𝜏.

In the following theorem we establish an integral expression of generalized

axial-symmetric potential that is a generalisation of integral expressions obtained

by A.G. Mackie [44], P. Henrici [45], Yu.P. Krivenkov [46] and G.N. Polozhii [47].

Theorem 4.12 ([48]). If 𝑚>0and 𝐹is an analytic function in the domain 𝐸𝑧,

then the function

𝑢(𝑥, 𝑦)= 1

2𝜋𝑖∣𝑦∣𝑚−1

Γ𝑧¯𝑧

𝐹(𝑡)(𝑡−𝑧)(𝑡−¯𝑧)𝑚/2−1𝑑𝑡 , 𝑧 =𝑥+𝑖𝑦, (4.28)

satisﬁes equation (4.25) on the set {(𝑥, 𝑦)∈𝐸:𝑦∕=0}. Moreover, there exists the

limit

lim

(𝑥,𝑦)→(𝑥0,0) 𝑢(𝑥, 𝑦)= B𝑚

2,1

2

2𝜋𝐹(𝑥0)∀𝑥0∈ℝ:(𝑥0,0) ∈𝐸,

where B(𝑝, 𝑞)is the Euler beta function.

We proved converse theorems on integral expressions of axial symmetric po-

tentials and Stokes ﬂow functions in domains of a meridian plane.

210 S.A. Plaksa

Theorem 4.13 ([32, 35]). Suppose that the axial-symmetric potential 𝜑(𝑥, 𝑟)is even

with respect to the variable 𝑟in the domain 𝐸. Then there exists the unique function

𝐹analytic in the domain 𝐸𝑧and satisfying the condition

𝐹(¯𝑧)=𝐹(𝑧)∀𝑧∈𝐷𝑧(4.29)

such that the equality (4.22) is fulﬁlled for all (𝑥, 𝑟)∈𝐸.

Theorem 4.14 ([34, 35]). Suppose that the Stokes ﬂow function 𝜓(𝑥, 𝑟)is even with

respect to the variable 𝑟in the domain 𝐸and satisﬁes the additional assumption

𝜓(𝑥, 0) ≡0∀(𝑥, 0) ∈𝐸. (4.30)

Then there exists a function 𝐹0analytic in the domain 𝐸𝑧such that the equality

(4.23) is fulﬁlled with 𝐹=𝐹0for all (𝑥, 𝑟)∈𝐸. Moreover, any analytic function

𝐹which satisﬁes the condition (4.29) and the equality (4.23) for all (𝑥, 𝑟)∈𝐷is

expressed in the form 𝐹(𝑧)=𝐹0(𝑧)+𝐶,where 𝐶is a real constant.

Note that the requirement (4.30) is natural. For example, for the model of

steady ﬂow of an ideal incompressible ﬂuid without sources and vortexes it means

that the axis 𝑂𝑥 is a line of ﬂow.

Using integral expressions (4.26) and (4.27), we developed a method for

eﬀectively solving boundary problems for axial-symmetric potential ﬁelds (see

[4, 32–36]).

5. Monogenic functions in the biharmonic algebra

5.1. Biharmonic algebra

We say that an associative commutative two-dimensional algebra 𝔹with the unit

1 over the ﬁeld ℂis biharmonic if in 𝔹there exists a biharmonic basis {𝑒1,𝑒

satisfying the conditions

(𝑒2

1+𝑒2

2)2=0,𝑒

1+𝑒2

2∕=0.(5.1)

V.F. Kovalev and I.P. Mel’nichenko [49] found a multiplication table for a

biharmonic basis {𝑒1,𝑒

2}:

𝑒1=1,𝑒

2=𝑒1+2𝑖𝑒2.(5.2)

In the paper [50] I.P. Mel’nichenko proved that there exists a unique bihar-

monic algebra 𝔹with a non-biharmonic basis {1,𝜌}, for which 𝜌2=0.Moreover,

he constructed all biharmonic bases in the form:

𝑒1=𝛼1+𝛼2𝜌, 𝑒

2=±𝑖𝛼1+𝛼2−1

2𝛼1𝜌,(5.3)

where complex numbers 𝛼1∕=0,𝛼2can be chosen arbitrarily. In particular, for the

basis (5.2) in the equalities (5.3) we choose 𝛼1=1,𝛼2= 0 and + of the double

sign:

𝑒1=1,𝑒

2=𝑖−𝑖

2𝜌, (5.4)

Commutative Algebras and Equations of Mathematical Physics 211

Note that every analytic function Φ(𝜁)ofthevariable𝜁=𝑥𝑒1+𝑦𝑒2satisﬁes

the two-dimensional biharmonic equation

(Δ2)2𝑈(𝑥, 𝑦):=∂4

∂𝑥4+2 ∂4

∂𝑥2∂𝑦2+∂4

∂𝑦4𝑈(𝑥, 𝑦) = 0 (5.5)

owing to the relations (5.1) and (Δ2)2Φ=Φ

(4)(𝜁)(𝑒2

1+𝑒2

2)2.

The algebra 𝔹has the unique maximum ideal ℐ:= {𝜆𝜌 :𝜆∈ℂ}which is

also the radical of 𝔹. In what follows, 𝑓:𝔹→ℂis the linear functional such that

the maximum ideal ℐis its kernel and 𝑓(1) = 1.

5.2. Monogenic functions given in a biharmonic plane.

Cauchy–Riemann conditions

Consider a biharmonic plane 𝜇𝑒1,𝑒2:= {𝜁=𝑥𝑒

1+𝑦𝑒

2:𝑥, 𝑦 ∈ℝ}which is a

linear envelope generated by the elements 𝑒1,𝑒

2of biharmonic basis (5.3). In what

follows, 𝜁=𝑥𝑒

1+𝑦𝑒

2and 𝑥, 𝑦 ∈ℝ.

Let 𝐺𝜁be a domain in the biharmonic plane 𝜇𝑒1,𝑒2. Inasmuch as divisors of

zero don’t belong to the plane 𝜇𝑒1,𝑒2, the Gateaux derivative of function Φ : 𝐺𝜁→

𝔹coincides with the derivative

Φ′(𝜁) := lim

ℎ→0,ℎ∈𝜇𝑒1,𝑒2Φ(𝜁+ℎ)−Φ(𝜁)ℎ−1.

Therefore, we deﬁne monogenic functions as functions Φ : 𝐺𝜁→𝔹for which the

derivative Φ′(𝜁) exists in every point 𝜁∈𝐺𝜁.

It is established in the paper [49] that a function Φ(𝜁) is monogenic in a

domain of biharmonic plane generated by the biharmonic basis (5.4) if and only if

the following Cauchy–Riemann condition is satisﬁed:

∂Φ(𝜁)

∂𝑦 =∂Φ(𝜁)

∂𝑥 𝑒2.(5.6)

It can similarly be proved that a function Φ : 𝐺𝜁→𝔹is monogenic in a

domain 𝐺𝜁of an arbitrary biharmonic plane 𝜇𝑒1,𝑒2if and only if the following

equality is fulﬁlled:

∂Φ(𝜁)

∂𝑦 𝑒1=∂Φ(𝜁)

∂𝑥 𝑒2∀𝜁=𝑥𝑒1+𝑦𝑒2∈𝐺𝜁.(5.7)

Below in Section 5, all stated results are obtained jointly with S.V. Gryshchuk

(see also [51]).

5.3. A constructive description of monogenic functions given in a biharmonic plane

Let 𝐷:= 𝑓(𝐺𝜁)and𝐴be the linear operator which assigns the function 𝐹:

𝐷→ℂto every function Φ : 𝐺𝜁→𝔹by the formula 𝐹(𝜉):=𝑓(Φ(𝜁)), where

𝜉:= 𝑓(𝜁)=𝛼1(𝑥±𝑖𝑦).

It is evident that if Φ is a monogenic function in the domain 𝐺𝜁,then𝐹is an

analytic function in the domain 𝐷, i.e., 𝐹is either holomorphic in the case where

𝜉=𝛼1(𝑥+𝑖𝑦) or antiholomorphic in the case where 𝜉=𝛼1(𝑥−𝑖𝑦).

212 S.A. Plaksa

The following theorem can be proved similarly to Theorem 2.3.

Theorem 5.1. Every monogenic function Φ:𝐺𝜁→𝔹canbeexpressedintheform

Φ(𝜁)= 1

2𝜋𝑖 

Γ𝜁

(𝐴Φ)(𝑡)(𝑡−𝜁)−1𝑑𝑡 +Φ

0(𝜁)∀𝜁∈𝐺𝜁,(5.8)

where Γ𝜁is an arbitrary closed rectiﬁable curve in 𝐷that embraces the point 𝑓(𝜁),

and Φ0:𝐺𝜁→ℐis a monogenic function taking values in the radical ℐ.

Note that the complex number 𝜉=𝑓(𝜁) is the spectrum of 𝜁∈𝔹,andthe

integral in the equality (5.8) is the principal extension (see [8, p. 165]) of analytic

function 𝐹(𝜉)=(𝐴Φ)(𝜉) of the complex variable 𝜉into the domain 𝐺𝜁.

It follows from Theorem 5.1 that the algebra of monogenic in 𝐺𝜁functions is

decomposed into the direct sum of the algebra of principal extensions of analytic

functions of a complex variable and the algebra of monogenic in 𝐺𝜁functions

taking values in the radical ℐ.

In the following theorem we describe all monogenic functions given in the

domain 𝐺𝜁and taking values in the radical ℐ.

Theorem 5.2. Every monogenic function Φ0:𝐺𝜁→ℐcanbeexpressedintheform

Φ0(𝜁)=𝐹0(𝜉)𝜌∀𝜁∈𝐺𝜁,(5.9)

where 𝐹0:𝐷→ℂis an analytic function and 𝜉=𝑓(𝜁).

Proof. Substituting the function (5.9) in the equality (5.7) in place of Φ and taking

into account the equalities 𝜌𝑒1=𝛼1𝜌,𝜌𝑒2=±𝛼1𝑖𝜌,weget

𝛼1

∂𝐹0(𝜉)

∂𝑦 𝜌=±𝛼1

∂𝐹0(𝜉)

∂𝑥 𝑖𝜌 ∀𝜉∈𝐷. (5.10)

From the equality (5.10), taking into account the uniqueness of decomposition

of elements of 𝔹with respect to the basis {1,𝜌}, we obtain the equality

∂𝐹0(𝜉)

∂𝑦 =±𝑖∂𝐹0(𝜉)

∂𝑥 ∀𝜉∈𝐷.

Thus, the function 𝐹0is either holomorphic in 𝐷in the case where 𝜉=

𝛼1(𝑥+𝑖𝑦) or antiholomorphic in 𝐷in the case where 𝜉=𝛼1(𝑥−𝑖𝑦), i.e., 𝐹0is

analytic in the domain 𝐷. The theorem is proved. □

It follows from the equalities (5.8), (5.9) that any monogenic function Φ :

𝐺𝜁→𝔹can be constructed by means of two complex analytic in 𝐷functions 𝐹,

𝐹0in the form:

Φ(𝜁)= 1

2𝜋𝑖 

Γ𝜁

𝐹(𝑡)(𝑡−𝜁)−1𝑑𝑡 +𝐹0(𝑓(𝜁))𝜌∀𝜁∈𝐺𝜁.(5.11)

Commutative Algebras and Equations of Mathematical Physics 213

Moreover, by using the expression

(𝑡−𝜁)−1=1

𝑡−𝜉−1

2𝛼1

2𝛼2𝜉±𝑖𝑦

(𝑡−𝜉)2𝜌,

∀𝜁=𝑥𝑒1+𝑦𝑒2∈𝐺𝜁∀𝑡∈ℂ:𝑡∕=𝜉=𝛼1(𝑥±𝑖𝑦),

the principal extension of an analytic in 𝐷function 𝐹into 𝐺𝜁can explicitly be

constructed in the form

2𝜋𝑖 

Γ𝜁

𝐹(𝑡)(𝑡−𝜁)−1𝑑𝑡 =𝐹(𝜉)−𝐹′(𝜉)

𝛼1𝛼2𝜉±𝑖𝑦

2𝜌,

𝜉=𝑓(𝜁)∈𝐷, ∀𝜁=𝑥𝑒1+𝑦𝑒2∈𝐺𝜁.

(5.12)

Note that in a particular case, in the paper [49] principal extensions of ana-

lytic functions of a complex variable was explicitly constructed into the biharmonic

plane generated by the biharmonic basis (5.4).

The following theorem can be proved similarly to Theorem 2.7.

Theorem 5.3. Every monogenic function Φ:𝐺𝜁→𝔹has derivatives of al l orders

in the domain 𝐺𝜁.

5.4. Isomorphism of algebras of monogenic functions

There is an isomorphism between algebras of monogenic functions at transition

from a biharmonic basis to another one. By ℳ(𝜇𝑒1,𝑒2,𝐺

𝜁) we denote the algebra

of monogenic functions in a domain 𝐺𝜁⊂𝜇𝑒1,𝑒2.

Theorem 5.4. Let {𝑒1,𝑒

2}be the biharmonic basis composed of the elements (5.4)

and {˜𝑒1,˜𝑒2}be an arbitrary biharmonic basis composed of elements of the form

(5.3).Let𝐺𝜁be a domain in the biharmonic plane 𝜇𝑒1,𝑒2and 

𝐺˜

𝜁:= {

𝜁=𝑥˜𝑒1±𝑦˜𝑒2:

𝜁=𝑥𝑒1+𝑦𝑒2∈𝐺𝜁}be the congruent domain in the biharmonic plane 𝜇˜𝑒1,˜𝑒2.Then

the algebras ℳ(𝜇𝑒1,𝑒2,𝐺

𝜁),ℳ(𝜇˜𝑒1,˜𝑒2,

𝐺˜

𝜁)are isomorphic, and the correspondence

ℳ(𝜇𝑒1,𝑒2,𝐺

𝜁)∋Φ←→ 

Φ∈ℳ(𝜇˜𝑒1,˜𝑒2,

𝐺˜

𝜁)are established by the equality



Φ(

𝜁)=Φ(𝜁)+Φ

′(𝜁)(𝑥𝑟1+𝑦𝑟2)𝜌,

where 𝑟1:= 𝛼2/𝛼1,𝑟2:= 𝑖(𝛼2

1+2𝛼1𝛼2−1)/(2𝛼2

1)and 𝛼1,𝛼2are the same com-

plex numbers which are situated in the equalities of the form (5.3) for elements of

the basis {˜𝑒1,˜𝑒2}.

5.5. A representation of a biharmonic function in the form of the ﬁrst component

of a monogenic function

In what follows, the basic elements 𝑒1,𝑒

2are deﬁned by the equalities (5.4) and

𝜁=𝑥𝑒

1+𝑦𝑒

2,𝑧=𝑥+𝑖𝑦 and 𝑥, 𝑦 ∈ℝ.

𝑈:𝐺→ℝis called a biharmonic function in a domain 𝐺⊂ℝ2if it satisﬁes

equation (5.5) in 𝐺.

214 S.A. Plaksa

We shall prove that every biharmonic function 𝑈1(𝑥, 𝑦) in a bounded simply

connected domain 𝐺⊂ℝ2is the ﬁrst component of some monogenic function

Φ(𝜁)=𝑈1(𝑥, 𝑦)𝑒1+𝑈2(𝑥, 𝑦)𝑖𝑒1+𝑈3(𝑥, 𝑦 )𝑒2+𝑈4(𝑥, 𝑦)𝑖𝑒2,(5.13)

in the corresponding domain 𝐺𝜁:= {𝜁=𝑥𝑒1+𝑦𝑒2:(𝑥, 𝑦)∈𝐺}of the biharmonic

plane 𝜇𝑒1,𝑒2,where𝑈𝑘:𝐺→ℝfor 𝑘= 1,4.

At ﬁrst, consider some auxiliary statements.

Lemma 5.5. Every monogenic function (5.13) with 𝑈1≡0is of the form

Φ(𝜁)=𝑖(−𝑎𝑥2+𝑘𝑥 −𝑎𝑦2−𝑏𝑦 +𝑛)+𝑒2(2𝑎𝑦2+2𝑏𝑦 +𝑐)

+𝑖𝑒2(−2𝑎𝑥𝑦 −𝑏𝑥 +𝑘𝑦 +𝑚)∀𝜁∈𝜇𝑒1,𝑒2,(5.14)

where 𝑎,𝑏,𝑐,𝑘,𝑚,𝑛 are arbitrary real constants.

To prove Lemma 5.5, taking into account the identity 𝑈1≡0, one should

integrate the Cauchy–Riemann condition (5.6) rewritten in expanded form:

0=∂𝑈3(𝑥, 𝑦)

∂𝑥 ,

∂𝑈2(𝑥, 𝑦)

∂𝑦 =∂𝑈4(𝑥, 𝑦)

∂𝑥 ,

∂𝑈3(𝑥, 𝑦)

∂𝑦 =−2∂𝑈4(𝑥, 𝑦)

∂𝑥 ,

∂𝑈4(𝑥, 𝑦)

∂𝑦 =∂𝑈2(𝑥, 𝑦)

∂𝑥 +2∂𝑈3(𝑥, 𝑦)

∂𝑥 .

Lemma 5.6. If 𝐹is a holomorphic function in a bounded simply connected domain

𝐷⊂ℂ, then the functions

Φ1(𝜁)=𝑢(𝑥, 𝑦)+𝑖𝑣(𝑥, 𝑦)−𝑒2𝑣(𝑥, 𝑦)+𝑖𝑒2𝑢(𝑥, 𝑦),

Φ2(𝜁)=𝑦𝑢(𝑥, 𝑦)+𝑖𝑦𝑣(𝑥, 𝑦)+𝑒2(𝒰(𝑥, 𝑦)−𝑦𝑣(𝑥, 𝑦))

+𝑖𝑒2(𝒱(𝑥, 𝑦)+𝑦𝑢(𝑥, 𝑦)) ,

Φ3(𝜁)=𝑥𝑢(𝑥, 𝑦)+𝑖𝑥𝑣(𝑥, 𝑦)+𝑒2(𝒱(𝑥, 𝑦)−𝑥𝑣(𝑥, 𝑦 ))

+𝑖𝑒2(𝑥𝑢(𝑥, 𝑦)−𝒰(𝑥, 𝑦)) ∀𝜁=𝑥𝑒1+𝑦𝑒2∈𝐺𝜁

are monogenic in the domain 𝐺𝜁≡{𝜁=𝑥𝑒1+𝑦𝑒2:𝑥+𝑖𝑦 ∈𝐷}of a biharmonic

plane 𝜇𝑒1,𝑒2,where

𝑢(𝑥, 𝑦):=Re𝐹(𝜉),𝑣(𝑥, 𝑦):=Im𝐹(𝜉),

𝒰(𝑥, 𝑦):=Reℱ(𝜉),𝒱(𝑥, 𝑦):=Imℱ(𝜉)∀𝜉∈𝐷

and ℱis a primitive function for the function 𝐹.

To prove Lemma 5.6, it is easy to show that the functions Φ1,Φ

2,Φ

3satisfy

the conditions of the form (5.6).

Commutative Algebras and Equations of Mathematical Physics 215

It is well known that every biharmonic function 𝑈1(𝑥, 𝑦) in the domain 𝐺is

expressed by the Goursat formula

𝑈1(𝑥, 𝑦)=Re(𝜑(𝜉)+¯

𝜉𝜓(𝜉)),𝜉=𝑥+𝑖𝑦, (5.15)

where 𝜑,𝜓are holomorphic functions in the domain 𝐷≡{𝑥+𝑖𝑦 :(𝑥, 𝑦)∈𝐺},

𝜉:= 𝑥−𝑖𝑦.

Theorem 5.7. Every biharmonic function 𝑈1(𝑥, 𝑦)in a bounded simp ly connected

domain 𝐺⊂ℝ2is the ﬁrst component in the decomposition (5.13) of the function

Φ(𝜁)=𝜑(𝜉)+¯

𝜉𝜓(𝜉)+𝑖𝑒2𝜑(𝜉)+¯

𝜉𝜓(𝜉)−2ℱ(𝜉),(5.16)

monogenic in the corresponding domain 𝐺𝜁of the biharmonic plane 𝜇𝑒1,𝑒2,where

𝜑,𝜓are the same functions as in the equality (5.15) and ℱis a primitive function

for the function 𝜓. Moreover, all monogenic in 𝐺𝜁functions for which the ﬁrst

component in the decomposition (5.13) is the given function 𝑈1are expressed as

the sum of the functions (5.14) and (5.16).

Proof. Introducing the functions

𝑢1(𝑥, 𝑦):=Re𝜑(𝑧),𝑢

2(𝑥, 𝑦):=Re𝜓(𝑧),𝑣

2(𝑥, 𝑦):=Im𝜓(𝑧),

we rewrite the equality (5.15) in the form

𝑈1(𝑥, 𝑦)=𝑢1(𝑥, 𝑦)+𝑥𝑢2(𝑥, 𝑦)+𝑦𝑣2(𝑥, 𝑦).(5.17)

Now, it follows from equality (5.17) and Lemma 5.6 that the function (5.16) is

monogenic in the domain 𝐺𝜁and its ﬁrst component in the decomposition (5.13)

is the given function 𝑈1. Finally, it evidently follows from Lemma 5.5 that all

monogenic in 𝐺𝜁functions for which the ﬁrst component in the decomposition

(5.13) is the given function 𝑈1are expressed as the sum of functions (5.14) and

(5.16). The theorem is proved. □

5.6. Integral theorems

In contrast to the papers [12, 13], where integral theorems are established for func-

tions diﬀerentiable in the sense of Lorch in domains of a commutative associative

Banach algebra, we establish similar results for monogenic functions Φ : 𝐺𝜁→𝔹

givenonlyinadomain𝐺𝜁of the biharmonic plane 𝜇𝑒1,𝑒2instead of a domain of

the whole algebra 𝔹. Moreover, note that the integral Cauchy formula established

in the papers [12, 13] is not applicable to a monogenic function Φ : 𝐺𝜁→𝔹be-

cause it deals with an integration along a curve on which the function Φ is not

given, generally speaking.

For the Euclidian norm ∥𝑎∥:= ∣𝜉1∣2+∣𝜉2∣2,where 𝑎=𝜉1𝑒1+𝜉2𝑒2and

𝜉1,𝜉

2∈ℂ, in the algebra 𝔹the inequality

∥𝑎𝑏∥≤√10 ∥𝑎∥∥𝑏∥∀𝑎,𝑏∈𝔹(5.18)

is fulﬁlled.

In the same way as in the complex plane, a rectiﬁable curve and an integral

along a rectiﬁable curve are deﬁned in the biharmonic plane 𝜇𝑒1,𝑒2.

216 S.A. Plaksa

The Cauchy integral theorem and integral formula for monogenic functions

of the variable 𝜁∈𝜇𝑒1,𝑒2are proved by a classic scheme (see, for example, [26])

by using the inequality (5.18). For a proof of the Cauchy integral formula we use

as well the equality (3.8) which is also fulﬁlled in the biharmonic plane 𝜇𝑒1,𝑒2.

Thus, the following statement is true:

Theorem 5.8. Suppose that the boundary ∂𝐺𝜁of domain 𝐺𝜁is a closed Jordan

rectiﬁable curve, and a function Φ: 𝐺𝜁−→ 𝔹is continuous in the closure 𝐺𝜁of

the domain 𝐺𝜁and is monogenic in 𝐺𝜁. Then the fol lowing equalities are fulﬁlled:



∂𝐺𝜁

Φ(𝜏)𝑑𝜏 =0 (the Cauchy theorem),(5.19)

Φ(𝜁)= 1

2𝜋𝑖 

∂𝐺𝜁

Φ(𝜏)(𝜏−𝜁)−1𝑑𝜏 ∀𝜁∈𝐺𝜁(the Cauchy formula).(5.20)

For functions of the biharmonic variable 𝜁, the following Morera theorem can

be established in the usual way (see, for example, [26]) by using Theorem 5.3 and

the inequality (5.18).

Theorem 5.9. If a function Φ: 𝐺𝜁−→ 𝔹is continuous in a domain 𝐺𝜁and satisﬁes

the equality (3.6) for every triangle △𝜁such that △𝜁⊂𝐺𝜁, then the function Φis

monogenic in the domain 𝐺𝜁.

5.7. The Taylor expansion

Consider a problem on an expansion of a monogenic in 𝐷𝜁function Φ in the Taylor

power series. Applying to the function (5.20) a method similar to a method for

expanding holomorphic functions, which is based on an expansion of the Cauchy

kernel in a power series (see, for example, [26, p. 107]), we obtain immediately the

following expansion of the function Φ in the power series:

Φ(𝜁)= ∞



𝑛=0

𝑏𝑛(𝜁−𝜁0)𝑛,(5.21)

where

𝑏𝑛=Φ(𝑛)(𝜁0)

𝑛!=1

2𝜋𝑖 

Φ(𝜏)(𝜏−𝜁0)−1𝑛+1 𝑑𝜏 , 𝑛 =0,1,...,

and Γ is an arbitrary closed Jordan rectiﬁable curve in 𝐺𝜁that embraces the point

𝜁0. But in such a way it can only be proved that the series (5.21) is convergent in

a disk 𝐾𝑟(𝜁0):={𝜁∈𝜇𝑒1,𝑒2:∥𝜁−𝜁0∥<𝑟}with a radius 𝑟whichislessthanthe

distance between 𝜁0and the boundary of domain 𝐺𝜁. It is connected with that

fact that the constant √10 can not be replaced by 1 in the inequality (5.18).

Nevertheless, taking into account the equality (5.12) in the plane 𝜇𝑒1,𝑒2gen-

erated by the basis (5.4) and using the equality (5.11) which is transformed now

Commutative Algebras and Equations of Mathematical Physics 217

into the form

Φ(𝜁)=𝐹(𝜉)𝑒1−𝑖𝑦

2𝐹′(𝜉)−𝐹0(𝜉)𝜌∀𝜁∈𝐺𝜁,(5.22)

we can prove the convergence of the series (5.21) in the disk 𝐾𝑅(𝜁0)withthe

radius 𝑅:= min

𝜏∈∂𝐷𝜁∥𝜏−𝜁0∥.

Theorem 5.10. If a function Φ: 𝐺𝜁−→ 𝔹is monogenic in a domain 𝐺𝜁and

𝜁0=𝑥0𝑒1+𝑦0𝑒2is an arbitrary point in 𝐺𝜁,thenΦis expressed in the disk

𝐾𝑅(𝜁0)as the sum of the convergent power series (5.21). In this case

𝑏𝑛=𝑐𝑛+𝑐(0)

𝑛−(𝑛+1)𝑖𝑦0

2𝑐𝑛+1𝜌,(5.23)

where 𝑐𝑛and 𝑐(0)

𝑛are coeﬃcients of the Taylor series

𝐹(𝜉)= ∞



𝑛=0

𝑐𝑛(𝜉−𝜉0)𝑛,𝐹

0(𝜉)= ∞



𝑛=0

𝑐(0)

𝑛(𝜉−𝜉0)𝑛,𝜉

0=𝑥0+𝑖𝑦0,(5.24)

for the functions 𝐹and 𝐹0included in the equality (5.22).

Proof. Inasmuch as in the equality (5.22) the functions 𝐹and 𝐹0are holomorphic

in the domain 𝐷:= {𝜉=𝑥+𝑖𝑦 :𝑥𝑒1+𝑦𝑒2∈𝐺𝜁}, the series (5.24) are absolutely

convergent in the disk {𝜉∈ℂ:∣𝜉−𝜉0∣<𝑅}. Then we rewrite the equality (5.22)

in the form

Φ(𝜁)=𝑐0+∞



𝑛=1

𝑐𝑛(𝜉−𝜉0)𝑛−𝑖(𝑦−𝑦0)

2𝑛(𝜉−𝜉0)𝑛−1𝜌

−𝑖𝑦0

∞



𝑛=0

(𝑛+1)𝑐𝑛+1(𝜉−𝜉0)𝑛𝜌+∞



𝑛=0

𝑐(0)

𝑛(𝜉−𝜉0)𝑛𝜌.

Now, using the relations

(𝜁−𝜁0)𝑛=(𝜉−𝜉0)𝑛−𝑛𝑖𝜌

2(𝜉−𝜉0)𝑛−1,(𝜁−𝜁0)𝑛𝜌=(𝜉−𝜉0)𝑛𝜌(5.25)

for all 𝜁∈𝜇𝑒1,𝑒2and 𝑛=0,1,..., we obtain the expression (5.21), where co-

eﬃcients are deﬁned by the equality (5.23) and the series (5.21) is absolutely

convergent in the disk 𝐾𝑅(𝜁0). The theorem is proved. □

Now, in the same way as for holomorphic functions of a complex variable

(see., for example, [26, p. 118]), we obtain the following uniqueness theorem for

monogenic functions of a biharmonic variable.

Theorem 5.11. If two monogenic in 𝐺𝜁functions coincide on a set which have at

least one limit point belonging to the domain 𝐺𝜁, then they are identically equal in

the whole domain 𝐺𝜁.

218 S.A. Plaksa

5.8. Diﬀerent equivalent deﬁnitions of monogenic functions

Thus, we obtain the following theorem which gives diﬀerent equivalent deﬁnitions

of monogenic functions in the biharmonic plane:

Theorem 5.12. AfunctionΦ: 𝐺𝜁−→ 𝔹is monogenic in the domain 𝐺𝜁if and

only if one of the following conditions is satisﬁed:

(I) the components 𝑈𝑘,𝑘= 1,4,oftheexpansion(5.13) of the function Φare dif-

ferentiable in the domain 𝐺and the condition (5.6) is satisﬁed in the domain

𝐺𝜁;

(II) the function Φis expressed in the form (5.22), where the pair of holomorphic

in 𝐷functions 𝐹and 𝐹0is unique;

(III) the function Φis continuous in 𝐺𝜁and satisﬁes the equality (3.6) for every

triangle △𝜁such that △𝜁⊂𝐺𝜁;

(IV) for every 𝜁0∈𝐺𝜁there exists a neighborhood, in which the function Φis

expressed as the sum of the power series (5.21).

5.9. The Laurent expansion

Consider Laurent series in the biharmonic plane. Set 𝐾𝑟,𝑅(𝜁0):={𝜁∈𝜇𝑒1,𝑒2:0≤

𝑟<∥𝜁−𝜁0∥<𝑅≤∞}.

Theorem 5.13. Every monogenic function Φ: 𝐾𝑟,𝑅 (𝜁0)−→ 𝔹is expressed in the

ring 𝐾𝑟,𝑅 (𝜁0)as the sum of the convergent series

Φ(𝜁)= ∞



𝑛=−∞

𝑏𝑛(𝜁−𝜁0)𝑛,(5.26)

where (𝜁−𝜁0)𝑛:= ((𝜁−𝜁0)−1)−𝑛for 𝑛=−1,−2,...,

𝑏𝑛=1

2𝜋𝑖 

Φ(𝜏)(𝜏−𝜁0)−𝑛−1𝑑𝜏, 𝑛 =0,±1,±2,..., (5.27)

and Γis an arbitrary closed Jordan rectiﬁable curve in 𝐾𝑟,𝑅 (𝜁0)that embraces the

point 𝜁0.

Proof. Inasmuch as in the equality (5.22) the functions 𝐹and 𝐹0are holomorphic

in the ring {𝜉∈ℂ:𝑟<∣𝜉−𝜉0∣<𝑅}with its center in the point 𝜉0=𝑥0+𝑖𝑦0,

they are expanded into Laurent series

𝐹(𝜉)= ∞



𝑛=−∞

𝑐𝑛(𝜉−𝜉0)𝑛,

𝐹0(𝜉)= ∞



𝑛=−∞

𝑐(0)

𝑛(𝜉−𝜉0)𝑛,

Commutative Algebras and Equations of Mathematical Physics 219

which are absolutely convergent in the mentioned ring. Then we rewrite the equal-

ity (5.22) in the form

Φ(𝜁)=𝑐0+∞



𝑛=1

𝑐𝑛(𝜉−𝜉0)𝑛−𝑖(𝑦−𝑦0)

2𝑛(𝜉−𝜉0)𝑛−1𝜌

−𝑖𝑦0

∞



𝑛=0

(𝑛+1)𝑐𝑛+1(𝜉−𝜉0)𝑛𝜌

+∞



𝑛=0

𝑐(0)

𝑛(𝜉−𝜉0)𝑛𝜌+𝑐−11

𝜉−𝜉0

+𝑖(𝑦−𝑦0)

2(𝜉−𝜉0)2𝜌

+𝑐(0)

−1

𝜉−𝜉0

𝜌+−2



𝑛=−∞

𝑐𝑛(𝜉−𝜉0)𝑛−𝑖(𝑦−𝑦0)

2𝑛(𝜉−𝜉0)𝑛−1𝜌

−𝑖𝑦0

−2



𝑛=−∞

(𝑛+1)𝑐𝑛+1(𝜉−𝜉0)𝑛𝜌+−2



𝑛=−∞

𝑐(0)

𝑛(𝜉−𝜉0)𝑛𝜌.

Further, using the equalities (5.25) for all 𝜁∈𝐾𝑟,𝑅(𝜁0)and𝑛=0,±1,±2,...,

we obtain the expansion of the function Φ in the series (5.26), where coeﬃcients

are deﬁned by the equalities (5.23), and, moreover, the series (5.26) is absolutely

convergent in the ring 𝐾𝑟,𝑅(𝜁0). Multiplying by (𝜁−𝜁0)−𝑛−1both parts of the

equality (5.26) and integrating then along the curve Γ, we obtain the formulas

(5.27) for coeﬃcients of the series (5.26). The theorem is proved. □

5.10. The classiﬁcation of isolated singular points of monogenic functions

in the biharmonic plane

It is an evident fact that every convergent in 𝐾𝑟,𝑅 (𝜁0) series of the form (5.26)

with coeﬃcients from 𝔹is the Laurent series of its sum. Terms of the series (5.26)

with nonnegative powers form its regular part, and terms with negative powers

form the principal part of the series (5.26).

Let us compactify the algebra 𝔹by means of addition of an inﬁnite point. Let

us agree that every sequence 𝑤𝑛:= 𝜉1,𝑛 𝑒1+𝜉2,𝑛𝑒2with 𝜉1,𝑛 ,𝜉

2,𝑛 ∈ℂconverges

to the inﬁnite point in the case, where at least one of the sequences 𝜉1,𝑛,𝜉2,𝑛

converges to inﬁnity in the extended complex plane.

Now, for a removable singular point and a pole and an essential singular point

of a function Φ which is monogenic in a pierced neighborhood 𝐾0,𝑟(𝜁0) of a point

𝜁0∈𝜇𝑒1,𝑒2, one can give the same deﬁnitions as for appropriate notions in the

complex plane (see, for example, [26]).

Moreover, an isolated singular point of monogenic function Φ(𝜁) of the bihar-

monic variable 𝜁has a relation with the form of Laurent expansion of this function.

More precisely, the following statement is true:

Theorem 5.14. If in a pierced neighbourhood 𝐾0,𝑟(𝜁0)of an isolated singular point

𝜁0∈𝜇𝑒1,𝑒2of a monogenic function Φ: 𝐾0,𝑅(𝜁0)−→ 𝔹, the principal part of the

Laurent series (5.26):

220 S.A. Plaksa

a) equals to zero, then 𝜁0is a removable singular point;

b) contains only a ﬁnite number of nonzero terms, then 𝜁0is a pole;

c) contains an inﬁnite number of nonzero terms, then 𝜁0is either a pole or an

essential singular point.

Indeed, it is evident that in the case a) the point 𝜉0is a removable singular

point for the functions 𝐹and 𝐹0from the equality (5.22). It is also evident that

in the case b) the point 𝜉0is a pole at least for one of the functions 𝐹,𝐹0,and

is not an essential singular point for these functions. Therefore, the point 𝜁0is

a removable singular point of the function Φ in the case a) and is a pole of this

function in the case b). In the case c) the point 𝜁0can be either a pole of the

function Φ (for example, in the case where the point 𝜉0is a pole of the function

𝐹and is an essential singular point of the function 𝐹0) or an essential singular

point of this function (for example, in the case where 𝐹≡0andthepoint𝜉0is

an essential singular point of the function 𝐹0).

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Index

analogue of the Gauss–Ostrogradsky

formula, 193

analogue of the Stokes formula, 193

axial-symmetric potential, 203

axial-symmetric potential ﬁeld, 202

biharmonic algebra, 210

biharmonic basis, 210

commutative associative Banach

algebra, 179

Hamilton’s quaternion, 179

harmonic algebra, 179, 181

harmonic basis, 183

harmonic functions, 178

harmonic triad, 179

harmonic vectors, 178

hypercomplex functions, 179

inﬁnite-dimensional commutative

associative Banach, 197

maximal ideal, 183

monogenic function, 180

potential function, 178

potential solenoid, 178

semisimple algebra, 181

Stokes ﬂow function, 203

three-dimensional harmonic algebra, 180

S.A. Plaksa

Institute of Mathematics of the

National Academy of Sciences of Ukraine

3, Tereshchenkivska Street

Kiev, Ukraine

e-mail: plaksa@imath.kiev.ua

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