-logarithm for slice regular functions
Amedeo Altavilla
Università degli Studi di Bari “Aldo Moro”, ItalyChiara de Fabritiis
Università Politecnica delle Marche, Ancona, Italy
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Abstract
In this paper, we study the (possible) solutions of the equation , where is a slice regular never vanishing function on a circular domain of the quaternions and is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function which satisfies is called a -logarithm of . We provide necessary and sufficient conditions, expressed in terms of the zero set of the “vector” part of , for the existence of a -logarithm of , under a natural topological condition on the domain . By this way, we prove an existence result if has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of are finite, the domain is either the unit ball, or , or (the solid torus obtained by circularization in of the disc contained in and centered in with radius ), and a further condition on the “real part” of is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of , in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.
Cite this article
Amedeo Altavilla, Chiara de Fabritiis, -logarithm for slice regular functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 34 (2023), no. 2, pp. 491–529
DOI 10.4171/RLM/1016