  After Gentili and Struppa introduced in 2006 the theory of quaternionic slice
regular function, the theory has focused on functions on the so-called slice
domains. The present work defines the class of speared domains, which is a
rather broad extension of the class of slice domains, and proves that the
theory is extremely interesting on speared domains. A Semi-global Extension
Theorem and a Semi-global Representation Formula are proven for slice regular
functions on speared domains: they generalize and strengthen some known local
properties of slice regular functions on slice domains. A proper subclass of
speared domains, called hinged domains, is defined and studied in detail. For
slice regular functions on a hinged domain, a Global Extension Theorem and a
Global Representation Formula are proven. The new results are based on a novel
approach: one can associate to each slice regular function
$f:\Omega\to\mathbb{H}$ a family of holomorphic stem functions and a family of
induced slice regular functions. As we tighten the hypotheses on $\Omega$ (from
an arbitrary quaternionic domain to a speared domain, to a hinged domain),
these families represent $f$ better and better and allow to prove increasingly
stronger results.
