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The Weierstrass factorization theorem for slice regular functions over the quaternions

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Abstract

The class of slice regular functions of a quaternionic variable has been recently introduced and is intensively studied, as a quaternionic analogue of the class of holomorphic functions. Unlike other classes of quaternionic functions, this one contains natural quaternionic polynomials and power series. Its study has already produced a rather rich theory having steady foundations and interesting applications. The main purpose of this article is to prove a Weierstrass factorization theorem for slice regular functions. This result holds in a formulation that reflects the peculiarities of the quaternionic setting and the structure of the zero set of such functions. Some preliminary material that we need to prove has its own independent interest, like the study of a quaternionic logarithm and the convergence of infinite products of quaternionic functions.

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Authors and Affiliations

  1. Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134, Firenze, Italy

    Graziano Gentili & Irene Vignozzi

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  1. Graziano Gentili
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  2. Irene Vignozzi
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Correspondence to Graziano Gentili.

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Gentili, G., Vignozzi, I. The Weierstrass factorization theorem for slice regular functions over the quaternions. Ann Glob Anal Geom 40, 435–466 (2011). https://doi.org/10.1007/s10455-011-9266-0

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  • DOI: https://doi.org/10.1007/s10455-011-9266-0

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