Dirac delta as a generalized holomorphic function
Sekar Nugraheni, Paolo Giordano
TL;DR
The paper tackles the challenge of defining a robust space of generalized holomorphic functions in the complex plane that accommodates derivatives of continuous maps and objects like the Dirac delta. By formulating a non-Archimedean extension of the complex field via Robinson–Colombeau numbers, it develops Generalized Smooth Functions and a framework for generalized holomorphy, establishing Cauchy–Riemann-type results, Goursat-type theorems, and stability under composition. It demonstrates intrinsic embeddings of compactly supported distributions, including the Dirac delta, and clarifies advantages over Colombeau theory, such as closure under nonlinear operations and broader domains. The work lays foundations for nonlinear distributional PDEs and a Cauchy–Kowalevski program in subsequent papers, with potential impact on analysis and mathematical physics by providing a flexible, rigorous language for non-Archimedean generalized holomorphicity.
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 differentiability implies smoothness, intrinsic embedding of compactly supported 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.