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Record Nr. |
UNINA9910735090403321 |
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Autore |
Plaksa Sergiy A |
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Titolo |
Monogenic Functions in Spaces with Commutative Multiplication and Applications / / by Sergiy A. Plaksa, Vitalii S. Shpakivskyi |
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Pubbl/distr/stampa |
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Cham : , : Springer Nature Switzerland : , : Imprint : Birkhäuser, , 2023 |
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ISBN |
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Edizione |
[1st ed. 2023.] |
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Descrizione fisica |
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1 online resource (548 pages) |
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Collana |
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Frontiers in Mathematics, , 1660-8054 |
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Altri autori (Persone) |
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Disciplina |
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Soggetti |
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Functions of complex variables |
Functions of a Complex Variable |
Àlgebra commutativa |
Problemes de contorn |
Llibres electrònics |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Nota di contenuto |
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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 -- References -- 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 -- References -- 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 -- References -- 5 Algebraic-Analytic Properties of Monogenic Functions in the Three-Dimensional Harmonic Algebra with Two-Dimensional Radical -- 5.1 Cauchy-Riemann Conditions for |
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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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 12 Monogenic |
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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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 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 -- References -- 17 Monogenic Functions in a Topological Vector Space Associated with Axial-Symmetric Potential Fields -- 17.1 |
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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 -- References -- 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. |
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Sommario/riassunto |
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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. |
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