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After their introduction in 2006, quaternionic slice regular functions havemostly been studied over domains that are symmetric with respect to the realaxis. 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 notaxially symmetric, partly correcting the hypotheses of some previouslypublished results. In particular, this work includes a Local RepresentationFormula valid without the symmetry hypothesis. Moreover, it determines a classof domains, called simple, having the following property: every slice regularfunction on a simple domain can be uniquely extended to the symmetriccompletion of its domain.

A local representation formula for quaternionic slice regular functions