*-products, *-exponential, *-logarithm: some peculiarities of slice regular functions on the quaternions

Abstract: Slice regular functions on quaternions were introduced in 2006 by Gentili and Struppa in order to generalize the notion of holomorphic functions on complex numbers (for an effective introduction you can refer to C. Stoppato's seminar (https://mediacentral.ucl.ac.uk/Play/59248/). The theory had a quick development in several directions by many authors, in this talk I will focus on three unexpected behaviours of these functions. The first aspect we deal with is the *-product, which is the analogous of pointwise product for holomorphic functions; in particular we give an interpretation of this operation via two operators which resemble the scalar product and the vector product on R^3. The second point we investigate is a suitable extension of the notion of exponential of a slice regular function, namely the *-exponential exp_*(f) (originally introduced by Colombo, Sabadini and Struppa); we will describe some of its features, especially with regard to the non-commutativity of the *-product and to its connections with *-sine and *-cosine. Lastly, we study the possible existence and uniqueness of a *-logarithm of a never vanishing slice regular function, both on slice and on product domains of the quaternions. We give some existence and non-existence results for *-logarithm of never-vanishing slice regular functions (according to the splitting in real and vectorial part) and an accurate description of the possible uniqueness of the *-logarithm. This is a joint work with Amedeo Altavilla (Università di Bari).

complex variablesdynamical systems

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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