Total differentiability and monogenicity for functions in algebras of order 4 (Q2027991)

There are basically two ways to generalize the theory of complex analysis to higher dimensions. The first way is to study several complex variables. The other is to study the analyticity in higher dimensional algebras. Both directions have been intensely studied for several decades. In the second case, one of the main problems is to generalize the notion of holomorphy to these algebras. In this paper, the authors first discuss the notions of analyticity in associative algebras with unit, i.e., the total differentiability and monogenicity. A nice relationship between the total derivability and the differentiability is given in Proposition 2.4. Then, the authors recall some basic tools in algebraic analysis and use them to study the properties of analytic functions in two algebras of dimension 4, i.e., Algebras LXXXI and LXXIX, which played a relevant role in the work of the Italian school, but have never been fully investigated. At the end of this paper, the authors raise an interesting question which may spur additional research: whether it is possible to envision a holomorphicity theory for bicomplex numbers that more closely resembles single variable theories