Holomorphicity of Slice-Regular Functions
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Complex Analysis and Operator Theory
Holomorphicity of Slice-Regular Functions Samuele Mongodi1 Received: 26 June 2019 / Accepted: 10 March 2020 © Springer Nature Switzerland AG 2020
Abstract The aim of this work is to show how a number of results about slice-regular functions follow flawlessly from the analogous properties of holomorphic functions. For this purpose, we provide a general strategy by which properties of holomorphic functions can be translated in the setting of slice-regular functions. As an example of application of our method, we study the relation between the zeroes of a slice-regular function and the values of the corresponding stem function, showing that a slice-regular function vanishes if and only if the corresponding stem function takes values in a given complex analytic subset of C4 . This allows us to recover in this setting a number of properties of the zeroes of holomorphic functions. We also discuss how our strategy can be adapted in some other contexts, like the study of the distribution of zeroes, of meromorphic functions, of representation formulas. Keywords Slice-regular functions · Stem functions · Zero set of slice-regular functions · Holomorphic functions Mathematics Subject Classification 30G35 · 16H99 · 32A30 · 30C15
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Introduction . . . . . . . . . . . General Setting . . . . . . . . . . Geometry of the Set Z . . . . . . Zeroes of Slice-Regular Functions -Products and Poles . . . . . . .
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Communicated by Irene Sabadini. This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, Michael Shapiro and Daniele Struppa.
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Samuele Mongodi [email protected] Dipartimento di Matematica, Politecnico di Milano, Via Bonardi, 9, 20133 Milan, Italy 0123456789().: V,-vol
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6 Integral Kernels . . . 7 Norms . . . . . . . . 8 Further Investigations References . . . . . . . .
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1 Introduction The theory of slice-regular functions of a quaternionic variable poses itself as a possible generalization of the theory of holomorphic functions of a complex variable; the definition of slice-regular functions was given by Gentili and Struppa in [12,13]. Given q ∈ H, we write q = x +vy with x, y ∈ R and
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