Quaternionic Functional Calculus

Abstract

The first section of this chapter collects the main results on the theory of slice regular functions. Similarly to what happens in the theory of regular functions in the sense of Cauchy–Fueter, whose results sometimes resemble the analogous results for monogenic functions, also for slice regular functions we have that some statements and their proofs mimic those we proved in Chapter 2. They are repeated here for the reader’s convenience, especially because the notation in the quaternionic case might be simpler. Note that the richer structure of quaternions allows results which are not necessarily true for s-monogenic functions. The results that are specific to the quaternionic case or those for which the proofs are significantly different or simpler will be followed by their proofs.