Calcolo differenziale e integrale medio per le funzioni di una variabile complessa

Abstract

Si sviluppa, per le funzioni (non necessariamente analitiche) di una variabile complessa, un analogo del classico Calcolo differenziale e integrale, basato su un'estensione delle nozioni di derivata e integrale complessi (derivata media, integrale medio).

An analogous of the elementary Calcolous of a real variable is outlined for functions (non analyric, possibly) of a complex variable. The start-point is a generalized definition of the complex derivate (mean derivative), that for $C^{1}$functions coincides with the formal operator $\delta/\delta^{z},$and an analogous extension of the complex integral (mean definite integral). The principal results of the real Calculus remain valid in $C$. For analytic functions, in particular, one obtains new proofs of some classical theorems (GOURSAT, MORERA).