Entire functions in one complex variable are extremely relevant in several
areas ranging from the study of convolution equations to special functions. An
analog of entire functions in the quaternionic setting can be defined in the
slice regular setting, a framework which includes polynomials and power series
of the quaternionic variable. In the first chapters of this work we introduce
and discuss the algebra and the analysis of slice regular functions. In
addition to offering a self-contained introduction to the theory of
slice-regular functions, these chapters also contain a few new results (for
example we complete the discussion on lower bounds for slice regular functions
initiated with the Ehrenpreis-Malgrange, by adding a brand new Cartan-type
theorem).
The core of the work is Chapter 5, where we study the growth of entire slice
regular functions, and we show how such growth is related to the coefficients
of the power series expansions that these functions have. It should be noted
that the proofs we offer are not simple reconstructions of the holomorphic
case. Indeed, the non-commutative setting creates a series of non-trivial
problems. Also the counting of the zeros is not trivial because of the presence
of spherical zeros which have infinite cardinality. We prove the analog of
Jensen and Carath\'eodory theorems in this setting
