Monogenic Functions in Spaces with Commutative Multiplication and Applications / by Sergiy A. Plaksa, Vitalii S. Shpakivskyi.

Format:

Book

Author/Creator:

Plaksa, Sergiy A.

Contributor:

Shpakivskyi, Vitalii S.

Series:

Frontiers in Mathematics, 1660-8054

Language:

English

Subjects (All):

Functions of complex variables.

Functions of a Complex Variable.

Local Subjects:

Functions of a Complex Variable.

Physical Description:

1 online resource (548 pages)

Edition:

1st ed. 2023.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Birkhäuser, 2023.

Summary:

This monograph develops a theory of continuous and differentiable functions, called monogenic functions, in the sense of Gateaux functions taking values in some vector spaces with commutative multiplication. The study of these monogenic functions in various commutative algebras leads to a discovery of new ways of solving boundary value problems in mathematical physics. The book consists of six parts: Part I presents some preliminary notions and introduces various concepts of differentiable mappings of vector spaces. Part II - V is devoted to the study of monogenic functions in various spaces with commutative multiplication, namely, three dimensional commutative algebras with two-dimensional radical, finite-dimensional commutative associative algebras, infinite-dimensional vector spaces associated with the three-dimensional Laplace equation and infinite-dimensional vector spaces associated with axial-symmetric potential fields. Part VI presents some boundary value problems for axial-symmetric potential fields and develops effective analytic methods of solving these boundary value problems with various applications in mathematical physics. Graduate students and researchers alike benefit from this book.

Contents:

Intro

Foreword

Preface

Contents

1 Introduction

References

Part I Differentiable Mapping in Vector Spaces

2 Differentiation in Vector Spaces

2.1 Fréchet Differential

2.2 Gâteaux Differential

2.2.1 Gâteaux Differential and its Properties

2.2.2 Examples

2.2.3 Relation to the Fréchet derivative

3 Monogenic Functions in Vector Spaces with Commutative Multiplication

3.1 Differentiability in the Sense of Lorch and in the Sense of Gâteaux in Commutative Banach Algebras

3.1.1 Differentiability in the Sense of Lorch

3.1.2 Principal Extensions of Holomorphic Functions of a Complex Variable

3.1.3 Differentiability in the Sense of Gâteaux

3.2 Monogenic Functions

3.3 Weakening of Monogeneity Conditions in the Complex Plane

Part II Monogenic Functions in a Three-Dimensional Commutative Algebra with Two-Dimensional Radical

4 Three-Dimensional Harmonic Algebra with Two-Dimensional Radical

4.1 Three-Dimensional Harmonic Algebra A3

4.2 Harmonic Bases in the Algebra A3

5 Algebraic-Analytic Properties of Monogenic Functions in the Three-Dimensional Harmonic Algebra with Two-Dimensional Radical

5.1 Cauchy-Riemann Conditions for Functions Taking Values in the Algebra A3

5.2 Principal Extensions of Holomorphic Functions of a Complex Variable into the Algebra A3

5.3 Constructive Description of Monogenic Functions Taking Values in the Algebra A3

5.3.1 Main Lemma

5.3.2 Main Results

5.3.3 Principal Corollaries

5.3.4 Monogenic Functions and Solutions of Three-Dimensional Laplace Equation

5.4 Isomorphism of Algebras of Monogenic Functions

6 Integral Theorems and Series in the Three-Dimensional Harmonic Algebra with Two-Dimensional Radical

6.1 Gauss-Ostrogradsky Formula and Cauchy Theorem for a Surface Integral.

6.1.1 Hyperholomorphic Functions

6.1.2 Gauss-Ostrogradsky Formula in the Algebra A3

6.1.3 Cauchy Theorem for a Surface Integral

6.2 Stokes Formula and Cauchy Theorem for a Curvilinear Integral

6.2.1 Stokes Formula in the Algebra A3

6.2.2 Cauchy Theorem for a Curvilinear Integral

6.3 Morera Theorem

6.4 Cauchy Integral Formula

6.5 Power Series and Monogenic Functions

6.5.1 Taylor Theorem for Monogenic Functions

6.5.2 Uniqueness Theorem for Monogenic Functions

6.6 Laurent Series and a Classification of Singular Points of Monogenic Functions

6.6.1 Laurent Theorem for Monogenic Functions

6.6.2 Classification of Singular Points of Monogenic Functions

6.7 Logarithmic Residue of Monogenic Functions

6.8 Different Equivalent Definitions of Monogenic Functions and Weakening of Monogeneity Conditions

6.8.1 Equivalent Definitions of Monogenic Functions

6.8.2 Weakening of the Continuity Condition for a Function

6.8.3 An Analog of the Menchov-Trokhimchuk Theorem

7 Hypercomplex Cauchy-Type Integral

7.1 On the Existence of Limiting Values of Cauchy-Type Integrals in the Complex Plane

7.2 Existence of Limiting Values of Hypercomplex Cauchy-Type Integrals on the Line of Integration

7.3 Existence of Limiting Values of Hypercomplex Cauchy-Type Integral on the Boundary of Domain

Part III Monogenic Functions in a Finite-Dimensional Commutative Associative Algebra

8 Constructive Description of Monogenic Functions in a Finite-Dimensional Commutative Algebra

8.1 Cartan Basis in a Finite-Dimensional Commutative Associative Algebra

8.2 Monogenic Functions in Special Subspaces of Algebra

8.3 Expansion of Resolvent

8.4 Constructive Description of Monogenic Functions

8.4.1 Auxiliary Results.

8.4.2 Representation of Monogenic Functions via Holomorphic Functions of Complex Variables

8.4.3 Principal Corollaries

8.5 Special Cases

8.5.1 A Case Where um+1=um+2=…=un

8.5.2 A Case Where Br, p0

8.5.3 A Case Where n=m

8.6 Some Relations Between Monogenic Functions and PDEs

9 Contour Integral Theorems for Monogenic Functions in a Finite-Dimensional Commutative Algebra

9.1 Cauchy Theorem and Morera Theorem for a Curvilinear Integral

9.2 Cauchy Integral Formula

9.2.1 Main Result

9.2.2 On a Constant λ

9.2.3 Taylor Theorem

9.3 Equivalent Definitions of Monogenic Functions

9.4 Brief Review of Some Other Results

10 Cauchy Theorem for a Surface Integral in a Finite-Dimensional Commutative Algebra

10.1 Surface Integrals on Quadrable Surfaces

10.2 Hyperholomorphic Functions and Auxiliary Statements

10.3 Cauchy Theorem for a Surface Integral

10.3.1 Main Result

10.3.2 Some Remarks

11 On Monogenic Functions Given in Various Commutative Algebras

11.1 Characteristic Equation

11.1.1 Characteristic Equation in Various Commutative Algebras

11.1.2 Linear Independence of Vectors 1,e"0365e2(u),e"0365e3(u)

11.2 Monogenic Functions Given in Various Commutative Algebras

12 Monogenic Functions on Extensions of a Commutative Algebra

12.1 Characteristic Equation in the Algebras An

12.2 Extensions of an Algebra and Their Properties

12.3 Monogenic Functions on Extensions of the Algebra An

13 Hypercomplex Method for Solving Linear Partial Differential Equations with Constant Coefficients

13.1 Monogenic Functions in Domains of a Special Space Ed

13.2 Families of Solutions of Linear Partial Differential Equations with Constant Coefficients

13.2.1 An Equation with Partial Derivatives of Highest Order Only.

13.2.2 An Arbitrary Linear Partial Differential Equation with Constant Coefficients

13.3 Families of Solutions Generated by a Sequence of Extensions {Eρn}n=1∞

13.3.1 Solutions of Linear Partial Differential Equations with Constant Coefficients

13.3.2 Solutions of Equations with Partial Derivatives of Highest Order Only

13.4 Examples

13.4.1 Solutions of One Hydrodynamic Equation

13.4.2 Solutions of the Three-Dimensional Laplace Equation

13.4.3 Solutions of the Wave Equation

13.4.4 Solutions of the Equation of Transverse Oscillation of an Elastic Rod

13.4.5 Solutions of the Generalized Biharmonic Equation

13.4.6 Solutions of the Two-Dimensional Helmholtz Equation

13.4.7 Final Remarks

Part IV Monogenic Functions in Infinite-Dimensional Vector Spaces Associated with the Three-Dimensional Laplace Equation

14 Description of Spatial Potential Fields by Means of Monogenic Functions in Infinite-Dimensional Spaces with Commutative Multiplication

14.1 Infinite-Dimensional Harmonic Algebra F

14.2 Monogenic Functions in the Algebra F

14.3 Monogenic Functions in the Topological Vector Space F"0365F Containing the Algebra F

14.4 Relation Between Monogenic Functions in F"0365F and Harmonic Vectors

14.4.1 Some Conditions for the Existence of Harmonic Vectors

14.4.2 Relation Between Monogenic Functions and Harmonic Vectors

14.4.3 Monogeneity of Gâteaux Derivatives

14.5 Infinite-Dimensional Harmonic Algebra G and Monogenic Functions in G

14.6 Monogenic Functions in the Topological Vector Space G"0365G Containing the Algebra G and Relation to Harmonic Vectors

15 Monogenic Functions in Complexified Infinite-Dimensional Spaces with Commutative Multiplication

15.1 Monogenic and Analytic Functions in the Algebra FC

15.2 Integral Theorems for Monogenic Functions in FC.

15.3 Monogenic Functions in the Topological Vector Space F"0365FC and Relation to Spatial Potentials

15.3.1 An Extension of Monogenic Functions in F"0365FC

15.3.2 Relations to Spatial Potentials

15.4 Integral Theorems for Monogenic Functions in F"0365FC

Part V Monogenic Functions in an Infinite-Dimensional Vector Space Associated with Axial-Symmetric Potential Fields

16 Monogenic Functions in an Infinite-Dimensional Commutative Banach Algebra Associated with Axial-Symmetric Potential Fields

16.1 Spatial Stationary Axial-Symmetric Potential Solenoidal Fields

16.2 Infinite-Dimensional Commutative Banach Algebra HC

16.3 Monogenic and Analytic Functions in ``Meridian'' Plane μ

16.4 Relation to Axial-Symmetric Potential Fields

16.4.1 Relation of Monogenic Functions in Proper Domains to Axial-Symmetric Potential Fields

16.4.2 Integral Expression for the Axial-Symmetric Potential in a Proper Domain

16.4.3 Integral Expression for Stokes' Flow Function in a Proper Domain

16.5 Relation to Elliptic Equations Degenerating on an Axis

16.6 Monogenic Functions and Principal Extensions of Holomorphic Functions into Three-Dimensional Linear Manifold M

16.7 Integral Theorems for Monogenic Functions Taking Values in the Algebra HC

17 Monogenic Functions in a Topological Vector Space Associated with Axial-Symmetric Potential Fields

17.1 Monogenic Functions Taking Values in a Topological Vector Space H"0365HC Containing the Algebra HC

17.2 Integral Theorems for Monogenic Functions Taking Valuesin H"0365HC

17.3 Relation to Axial-Symmetric Potential Fields

Part VI Boundary Value Problems for Axial-Symmetric Potential Fields

18 Integral Representations for the Axial-Symmetric Potential and Stokes' Flow Function in an Arbitrary Simply-Connected Domain.

18.1 Direct Theorems.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

3-031-32254-1

OCLC:

1390921610