Holomorphic Extensions of Complex Probability Measures: From Fourier-Stieltjes Transforms to Riemann Surface Applications

This systematic and complete work carefully examines the holomorphic extensions of complex probability measures through the perceptive lens of Fourier-Stieltjes transforms and their crucial and deep applications to the complex theory of Riemann surfaces. We derive a system of rigorous theoretical results that are vital for extending complex-valued probability measures from their natural and originally specified domains using the analytic structure that resides intrinsically with their characteristic functions. Our complete and careful treatment covers fundamental existence and uniqueness theorems, also certain convergence results and explicit construction techniques that are needed for the effective creation of holomorphic extensions. We illustrate the intimate and significant relations that exist among complex probability theory and the geometry of Riemann surfaces and show how multi-valued probability functions may actually produce single-valued functions considered on properly crafted Riemann surfaces. Complete and careful proofs of all of our significant results are shown in the paper, including certain extension theorems of a new and original nature and complete and definitive characterizations of singularity structures as well as numerous and miscellaneous applications to quantum probability theory. Using specific and illustrative computational algorithms and careful and specific illustrative examples, we outline the intricate and real-world utility of said extensions across a wide variety of disciplines ranging from mathematical physics and signal processing and demonstrate the compelling research promise of our findings towards unlocking breakthroughs and innovations in pure and applied mathematics.