Regular Functions of a Quaternionic Variable

Description

This book surveys the foundations of the theory of slice regular functions over the quaternions, introduced in 2006, and gives an overview of its generalizations and applications. As in the case of other interesting quaternionic function theories, the original motivations were the richness of the theory of holomorphic functions of one complex variable and the fact that quaternions form the only associative real division algebra with a finite dimension n>2. (Slice) regular functions quickly showed particularly appealing features and developed into a full-fledged theory, while finding applications to outstanding problems from other areas of mathematics. For instance, this class of functions includes polynomials and power series. The nature of the zero sets of regular functions is particularly interesting and strictly linked to an articulate algebraic structure, which allows several types of series expansion and the study of singularities. Integral representation formulas enrich the theory and are fundamental to the construction of a noncommutative functional calculus. Regular functions have a particularly nice differential topology and are useful tools for the construction and classification of quaternionic orthogonal complex structures, where they compensate for the scarcity of conformal maps in dimension four. This second, expanded edition additionally covers a new branch of the theory: the study of regular functions whose domains are not axially symmetric. The volume is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general. Preface Preface to the Second Edition Introduction Contents 1 Definitions and Basic Results 1.1 Regular Functions 1.2 Affine Representation 1.3 Extension Results and Local Representation 1.4 Algebraic Structure Bibliographic Notes 2 Regular Power Series 2.1 The Distance σ 2.2 Convergence of Power Series Centered at p 2.3 Series Expansion at p and Analyticity Bibliographic Notes 3 Zeros 3.1 Basic Properties of the Zeros 3.2 Algebraic Properties of the Zeros 3.3 Topological Properties of the Zeros 3.4 On the Roots of Quaternions 3.5 Factorization of Polynomials 3.6 Multiplicity 3.7 Division Algorithm and Bezout Theorem 3.8 Gröbner Bases for Quaternionic Polynomials Bibliographic Notes 4 Infinite Products 4.1 Infinite Products of Quaternions 4.2 The Principal Branch of Quaternionic Logarithm 4.3 Infinite Products of Functions Defined on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper H) /StPNE pdfmark [/StBMC pdfmarkHps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 4.4 Convergence of an Infinite-Product 4.5 Convergence-Producing Regular Factors 4.6 Weierstrass Factorization Theorem Bibliographic Notes 5 Singularities 5.1 Regular Reciprocal and Quotients 5.2 Laurent Series and Expansion 5.3 Classification of Singularities 5.4 Poles and Quotients 5.5 Casorati–Weierstrass Theorem Bibliographic Notes 6 Integral Representations 6.1 Cauchy Theorem and Morera Theorem 6.2 Cauchy Integral Formula 6.3 Pompeiu Formula 6.4 Derivatives Using the Cauchy Formula 6.5 Coefficients of the Laurent Series Expansion 6.6 Argument Principle Bibliographic Notes 7 Maximum Modulus Theorem and Applications 7.1 Maximum and Minimum Modulus 7.2 Open Mapping Theorem 7.3 Real Parts of Regular Functions 7.4 Phragmén–Lindelöf Principles 7.5 An Ehrenpreis–Malgrange Lemma Bibliographic Notes 8 Spherical Series and Differential 8.1 Spherical Series and Expansions 8.2 Integral Formulas and Cauchy Estimates 8.3 Symmetric Analyticity 8.4 Differentiating Regular Functions 8.5 Rank of the Differential Bibliographic Notes 9 Fractional Transformations and the Unit Ball 9.1 Transformations of the Quaternionic Space 9.2 Regular Fractional Transformations 9.3 Transformations of the Quaternionic Riemann Sphere 9.4 Schwarz Lemma and Transformations of the Unit Ball 9.5 Rigidity and a Boundary Schwarz Lemma 9.6 Borel–Carathéodory Theorem 9.7 Bohr Theorem Bibliographic Notes 10 Generalizations 10.1 Direct Generalizations to Algebras Other Than ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper H) /StPNE pdfmark [/StBMC pdfmarkHps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 10.1.1 The Case of Octonions 10.1.2 The Case of ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R 3) /StPNE pdfmark [/StBMC pdfmarkR3ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark 10.1.3 The Slice Monogenic Case 10.2 Slice Functions Over ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper H) /StPNE pdfmark [/StBMC pdfmarkHps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and Other Alternative Real Algebras 10.3 An Alternative Approach to Slice Regularity 11 Function Theory Over Non-symmetric Slice Domains 11.1 Spherical Value and Derivative 11.2 Locally Slice Functions and their Algebraic Structure 11.3 Zero Sets 11.4 Quotients 11.5 Factorization of Zeros 11.6 Applications of Factorization 11.7 Semiregular Functions 11.8 Minimum Modulus Principle and Open Mapping Theorem 11.9 Integral Representation 11.10 Spherical Series Expansions Bibliographic Notes 12 Applications 12.1 Applications in Functional Analysis 12.1.1 Quaternionic Functional Calculus 12.1.2 Some Quaternionic Function Spaces 12.2 Applications in Differential Geometry 12.2.1 Orthogonal Complex Structures Induced by Regular Functions 12.2.2 A Direct Approach to Quaternionic Manifolds 12.3 Applications in Spatial Kinematics 12.3.1 Rational Rotation-Minimizing Frame Curves 12.3.2 Motion Polynomials over Dual Quaternions References Index