-logarithm for slice regular functions

  • Amedeo Altavilla

    Università degli Studi di Bari “Aldo Moro”, Italy
  • Chiara de Fabritiis

    Università Politecnica delle Marche, Ancona, Italy
$*$-logarithm for slice regular functions cover
Download PDF

This article is published open access under our Subscribe to Open model.


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