Singularities of slice regular functions

Beginning in 2006, G. Gentili and D.C. Struppa developed a theory of regular quaternionic functions with properties that recall classical results in complex analysis. For instance, in each Euclidean ball centered at 0 the set of regular functions coincides with that of quaternionic power series converging in the same ball. In 2009 the author proposed a classification of singularities of regular functions as removable, essential or as poles and studied poles by constructing the ring of quotients. In that article, not only the statements, but also the proving techniques were confined to the special case of balls centered at 0. In a subsequent paper, F. Colombo, G. Gentili, I. Sabadini and D.C. Struppa (2009) identified a larger class of domains, on which the theory of regular functions is natural and not limited to quaternionic power series. The present article studies singularities in this new context, beginning with the construction of the ring of quotients and of Laurent-type expansions at points other than the origin. These expansions, which differ significantly from their complex analogs, allow a classification of singularities that is consistent with the one given in 2009. Poles are studied, as well as essential singularities, for which a version of the Casorati-Weierstrass Theorem is proven.Comment: 25 pages, 1 figur

A local representation formula for quaternionic slice regular functions

After their introduction in 2006, quaternionic slice regular functions have mostly been studied over domains that are symmetric with respect to the real axis. This choice was motivated by some foundational results published in 2009, such as the Representation Formula for axially symmetric domains. The present work studies slice regular functions over domains that are not axially symmetric, partly correcting the hypotheses of some previously published results. In particular, this work includes a Local Representation Formula valid without the symmetry hypothesis. Moreover, it determines a class of domains, called simple, having the following property: every slice regular function on a simple domain can be uniquely extended to the symmetric completion of its domain.Comment: 10 pages, to appear in Proc. Amer. Math. So

A new series expansion for slice regular functions

A promising theory of quaternion-valued functions of one quaternionic variable, now called slice regular functions, has been introduced in 2006. The basic examples of slice regular functions are power series centered at 0 on their balls of convergence. Conversely, if f is a slice regular function then it admits at each point of its domain an expansion into power series, where the powers are taken with respect to an appropriately defined multiplication *. However, the information provided by such an expansion is somewhat limited by a fact: if the center p of the series does not lie on the real axis then the set of convergence needs not be a Euclidean neighborhood of p. We are now able to construct a new type of expansion that is not affected by this phenomenon: an expansion into series of polynomials valid in open subsets of the domain. Along with this construction, we present applications to the computation of the multiplicities of zeros and of partial derivatives.Comment: 20 pages, 1 figur

A unified notion of regularity in one hypercomplex variable

We define a very general notion of regularity for functions taking values in an alternative real $*$-algebra. Over Clifford numbers, this notion subsumes the well-established notions of monogenic function and slice-monogenic function. Over quaternions, in addition to subsuming the notions of Fueter-regular function and of slice-regular function, it gives rise to an entirely new theory, which we develop in some detail.Comment: 16 page

Completeness on the worm domain and the M\"untz-Sz\'asz problem for the Bergman space

In this paper we are concerned with the problem of completeness in the Bergman space of the worm domain $\mathcal{W}_\mu$ and its truncated version $\mathcal{W}'_\mu$. We determine some orthogonal systems and show that they are not complete, while showing that the union of two particular of such systems is complete. In order to prove our completeness result we introduce the Muentz-Szasz problem for the 1-dimensional Bergman space of the disk $\{\zeta : |\zeta-1|<1\}$ and find a sufficient condition for its solution.Comment: 14 pages, Author Accepted Manuscrip