Complex Slowly Varying Function

Title: Complex Slowly Varying Functions

Abstract: This paper establishes a rigorous theoretical framework for complex slowly varying functions, analogous in spirit to Karamata’s real-variable theory while incorporating genuinely complex-analytic phenomena. We formalize the foundational notions of Globally Slowly Varying (GSV), Sectorially Slowly Varying (SSV), Analytic Slow Variation (ASV), and Raywise Sectorially Slowly Varying (RSSV) functions. The project explores the dual tension between analytic flexibility and rigidity, proving the Karamata-Liouville Rigidity Theorem—which dictates that global slow variation forces non-vanishing entire functions to be absolute constants—and utilizing sectorial "pressure valves" to harbor complex dipole distributions. Furthermore, we construct a canonical infinite dipole product satisfying ASV and SSV via rigorous continuous majorization, and establish the fundamental ASV–SSV Equivalence Theorem via normal families and Cauchy's integral formula.