Entire Slice Regular Functions (springerbriefs In Mathematics)

This Briefs volume develops the theory of entire slice regular functions. It is the first self-contained, monographic work on the subject, offering all the necessary background information and detailed studies on several central topics, including estimates on the minimum modulus of regular functions, relations between Taylor coefficients and the growth of entire functions, density of their zeros, and the universality properties. The proofs presented here shed new light on the nature of the quaternionic setting and provide inspiration for further research directions. Also featuring an exhaustive reference list, the book offers a valuable resource for graduate students, postgraduate students and researchers in various areas of mathematical analysis, in particular hypercomplex analysis and approximation theory. Contents 5 1 Introduction 6 2 Slice Regular Functions: Algebra 11 2.1 Definition and Main Results 11 2.2 The Representation Formula 15 2.3 Multiplication of Slice Regular Functions 21 2.4 Quaternionic Intrinsic Functions 26 2.5 Composition of Power Series 28 3 Slice Regular Functions: Analysis 35 3.1 Some Integral Formulas 35 3.2 Riemann Mapping Theorem 42 3.3 Zeros of Slice Regular Functions 44 3.4 Modulus of a Slice Regular Function and Ehrenpreis--Malgrange Lemma 49 3.5 Cartan Theorem 56 4 Slice Regular Infinite Products 59 4.1 Infinite Products of Quaternions 59 4.2 Infinite Products of Functions 63 4.3 Weierstrass Factorization Theorem 68 4.4 Blaschke Products 77 5 Growth of Entire Slice Regular Functions 81 5.1 Growth Scale 81 5.2 Jensen Theorem 88 5.3 Carathéodory Theorem 93 5.4 Growth of the -Product of Entire Slice Regular Functions 102 5.5 Almost Universal Entire Functions 105 5.6 Entire Slice Regular Functions of Exponential Type 110 References 113 Index 120 Front Matter....Pages i-v Introduction....Pages 1-5 Slice Regular Functions: Algebra....Pages 7-30 Slice Regular Functions: Analysis....Pages 31-54 Slice Regular Infinite Products....Pages 55-76 Growth of Entire Slice Regular Functions....Pages 77-108 Back Matter....Pages 109-118