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Slice analysis is a generalization of the theory of holomorphic functions ofone complex variable to quaternions.  Among the new phenomena which appear in this context, there is the fact thatthe convergence domain of $f(q)=\Sigma_{n\in\mathbb{N}}(q-p)^{*n} a_n$, givenby 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 anatural topology for slice regular functions. It turns out that the naturaltopology is the so-called slice topology, which is different from the Euclideantopology and nicely adapts to the slice structure of quaternions. We extend thefunction theory of slice regular functions to any domains in the slicetopology.  Many fundamental results in the classical slice analysis for axiallysymmetric domains fail in our general setting.  We can even construct a counterexample to show that a slice regular functionin a domain cannot be extended to an axially symmetric domain.  In order to provide positive results we need to consider so-called path-slicefunctions instead of slice functions.  Along this line, we can establish an extension theorem and a representationformula in a slice-domain.

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