Extension theorem and representation formula in non-axially-symmetric domains for slice regular functions

Abstract

Slice analysis is a generalization of the theory of holomorphic functions of one complex variable to quaternions. Among the new phenomena which appear in this context, there is the fact that the convergence domain of $f(q)={\sum }_{n\in \mathbb{N}}(q-p{)}^{\ast n}{a}_n$, given by a $\sigma$-ball $\Sigma (p,r)$, is not open in $\mathbb{H}$ unless $p\in \mathbb{R}$. This motivates us to investigate, in this article, what is a natural topology for slice regular functions. It turns out that the natural topology is the so-called slice topology, which is different from the Euclidean topology and nicely adapts to the slice structure of quaternions. We extend the function theory of slice regular functions to any domains in the slice topology. Many fundamental results in the classical slice analysis for axially symmetric domains fail in our general setting. We can even construct a counterexample to show that a slice regular function in a domain cannot be extended to an axially symmetric domain. In order to provide positive results we need to consider so-called path-slice functions instead of slice functions. Along these lines, we can establish an extension theorem and a representation formula in a slice domain.