Dirac delta as a generalized holomorphic function

Abstract

The definition of a non-trivial space of generalized functions of a complex variable allowing to consider derivatives of continuous functions is a non-obvious task, e.g. because of Morera theorem, because distributional Cauchy–Riemann equations implies holomorphicity and of course because including Dirac delta seems incompatible with the identity theorem. Surprisingly, these results can be achieved if we consider a suitable non-Archimedean extension of the complex field, i.e. a ring where infinitesimal and infinite numbers return to be available. In this first paper, we set the definition of generalized holomorphic function and prove the extension of several classical theorems, such as Cauchy–Riemann equations, Goursat, Looman–Menchoff and Montel theorems, generalized complex differentiability implies smoothness, embedding of distributions, closure with respect to composition and hence non-linear operations on these generalized functions. The theory hence addresses several limitations of Colombeau theory of generalized holomorphic functions. The final aim of this series of papers is to prove the Cauchy–Kowalevski theorem including also distributional PDE or singular boundary conditions and nonlinear operations.