Generalized Holomorphic Functions: Sketches of a New Theory

We start from the simple idea to define a generalized holomorphic function (GHF) without already assuming the Cauchy-Riemann equations but using a natural definition of complex differentiability, i.e. of limit of the incremental ratio. The setting is that of Robinson-Colombeau ring, which generalizes the ring of Colombeau generalized numbers by considering a more general “growth condition” \((\rho _{\varepsilon })\) (called gauge) instead of the usual \(\rho _{\varepsilon }=\varepsilon \) . This natural definition actually uses two gauges in order to define two different sharp topologies on the domain and on the codomain, and hence a new notion of limit and little-oh in the definition of GHF. This conduct us to a more general theory, where several classical theorems of differential calculus can be extended from the ordinary holomorphic case to generalized holomorphic framework, overpassing several drawbacks of Colombeau theory of holomorphic functions.