A unified theory of regular functions of a hypercomplex variable

This work proposes a unified theory of regularity in one hypercomplex variable: the theory of T-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises Fueter regular functions, slice-regular functions and a recently-discovered function class. In the special case of Clifford-valued functions of one paravector variable, it encompasses monogenic functions, slice-monogenic functions, generalized partial-slice monogenic functions, and a variety of function classes not yet considered in literature. For T-regular functions over an associative *-algebra, this work provides integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle and a Representation Formula. It also proves some foundational results about T-regular functions over an alternative but nonassociative *-algebra, such as the real algebra of octonions.