ANALYTIC CONTINUATION OF COMPLEX MEASURES: A RIEMANN SURFACE PERSPECTIVE

Abstract

This systematic and complete work presents a rigorous theoretical framework for the analytic continuation of complex probability measures, leveraging the intrinsic analytic structure embedded within their Fourier-Stieltjes transforms. The theory of holomorphic extension, a cornerstone of complex analysis, is here applied to generalize complex-valued measures from their original real domains into the complex plane. We establish a comprehensive system of fundamental results, including necessary and sufficient conditions for the existence and uniqueness of these continuations, providing both convergence theorems and explicit construction techniques essential for their effective realization. This treatment includes the derivation of novel extension theorems and a definitive characterization of the singularity structures—such as branch points, poles, and essential singularities—that arise in the complex domain.