Book: Gentili et al – Regular Functions of a Quaternionic Variable


Regular Functions of a Quaternionic Variable Regular Functions of a Quaternionic Variable
Gentili, Graziano, Stoppato, Caterina, Struppa, Daniele C, Series: Springer Monographs in Mathematics, 2013, XIX, 185 p. 4 illus., 3 illus. in color, Ebook: ISBN 978-3-642-33871-7, 71.39 Euros, Hardcover: ISBN 978-3-642-33870-0, 84.95 Euros.

  • The book is entirely devoted to a new theory
  • Presents a state of the art survey of the theory of slice regular functions ​
  • The theory presented in the book is the basis for the solution to an outstanding problem, the construction of functional calculus in non commutative settings

The theory of slice regular functions over quaternions is the central subject of the present volume. This recent theory has expanded rapidly, producing a variety of new results that have caught the attention of the international research community. At the same time, the theory has already developed sturdy foundations. The richness of the theory of the holomorphic functions of one complex variable and its wide variety of applications are a strong motivation for the study of its analogs in higher dimensions. In this respect, the four-dimensional case is particularly interesting due to its relevance in physics and its algebraic properties, as the quaternion forms the only associative real division algebra with a finite dimension n>2. Among other interesting function theories introduced in the quaternionic setting, that of (slice) regular functions shows particularly appealing features. For instance, this class of functions naturally includes polynomials and power series. The zero set of a slice regular function has an interesting structure, strictly linked to a multiplicative operation, and it allows the study of singularities. Integral representation formulas enrich the theory and they are a fundamental tool for one of the applications, the construction of a noncommutative functional calculus.

The volume presents a state-of-the-art survey of the theory and a brief overview of its generalizations and applications. It is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general. ​

Keywords » 30G35, 30B10, 30C15, 30E20, 30C80 – Schwarz’s lemma – functions of hypercomplex variables and generalized variables – maximum principle – power series – zeros of polynomials

Contents
1 Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Regular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Affine Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Extension Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Algebraic Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Regular Power Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 The Distance  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Convergence of Power Series Centered at p . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Series Expansion at p and Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Basic Properties of the Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Algebraic Properties of the Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Topological Properties of the Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.4 On the Roots of Quaternions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Factorization of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7 Division Algorithm and Bezout Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 Gr¨obner Bases for Quaternionic Polynomials . . . . . . . . . . . . . . . . . . . . . . 46
4 Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Infinite Products of Quaternions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 The Quaternionic Logarithm.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Infinite Products of Functions Defined on H. . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Convergence of an Infinite -Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Convergence-Producing Regular Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6 Weierstrass Factorization Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1 Regular Reciprocal and Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Laurent Series and Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Classification of Singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Poles and Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.5 Casorati–Weierstrass Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Integral Representations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.1 Cauchy Theorem and Morera Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Cauchy Integral Formula.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3 Pompeiu Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.4 Derivatives Using the Cauchy Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.5 Coefficients of the Laurent Series Expansion .. . . . . . . . . . . . . . . . . . . . . . 96
6.6 Argument Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7 Maximum Modulus Theorem and Applications . . . . . . . . . . . . . . . . . . . . . . . . 103
7.1 Maximum and Minimum Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.3 Real Parts of Regular Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.4 Phragm´en–Lindel¨of Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.5 An Ehrenpreis–Malgrange Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8 Spherical Series and Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.1 Spherical Series and Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2 Integral Formulas and Cauchy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.3 Symmetric Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
8.4 Differentiating Regular Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.5 Rank of the Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
9 Fractional Transformations and the Unit Ball . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9.1 Transformations of the Quaternionic Space. . . . . . . . . . . . . . . . . . . . . . . . . 141
9.2 Regular Fractional Transformations .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9.3 Transformations of the Quaternionic Riemann Sphere .. . . . . . . . . . . . 145
9.4 Schwarz Lemma and Transformations of the Unit Ball . . . . . . . . . . . . 147
9.5 Rigidity and a Boundary Schwarz Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . 150
9.6 Borel–Carath´eodory Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
9.7 Bohr Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10 Generalizations and Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.1 Slice Regularity in Algebras Other than H. . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.1.1 The Case of Octonions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.1.2 The Case of R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.1.3 The Monogenic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.2 Quaternionic Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10.3 Orthogonal Complex Structures Induced by Regular Functions .. . 174

Source: http://www.springer.com/mathematics/analysis/book/978-3-642-33870-0

Leave a comment

Filed under publications

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s