Complex Slowly Varying Functions

Title: Complex Slowly Varying Functions

Abstract: Classical Karamata theory formalizes asymptotic scale invariance on the positive real
line under the minimal baseline of measurability. Past extensions to the complex plane
have largely circumvented the topological foundation by imposing a priori holomorphy
and vanishing logarithmic derivatives (𝑧𝑓′/𝑓 → 0), reducing scale invariance to a trivial
consequence of differential decay. In this paper, we establish an axiomatic foundation
for Complex Slowly Varying (CSV) functions by decoupling macroscopic scaling from
analyticity. Operating on interior buffered sectors Ω𝜂 under the minimal baseline of
continuous (𝐶
0
) unimodular fields, we prove the Complex Uniform Convergence Theo-
rem via the Baire Category Theorem and a Banach-space cocycle identity. We derive the
Generalized Karamata Representation Theorem for Continuous Sectorial Slow Variation
(SSV) using designated radial de Haan moving averages, isolating non-differentiable
fluctuations into an asymptotically flat pre-factor 𝑐(𝑧) → 1.
We show that isotropic Global Slow Variation (GSV) forces entire non-vanishing func-
tions to collapse into constants via Liouville rigidity, dictating that non-trivial complex
scale invariance structurally requires singularity harbors (the geometric pressure
valve). Within interior buffered sectors, we prove the fundamental Equivalence The-
orem (SSV ⟺ ASV) via a spatial Cauchy integral average, demonstrating that
holomorphy acts as an elliptic regularity clamp that extinguishes transverse phase
shearing, collapses the continuous representation to its path-independent canonical
form, unlocks complex scalar rotations (𝜆 ∈ ℂ∗
), and grants domain self-healing across
one-dimensional seams. Finally, we provide a constructive engine via the Primary
Dipole Theorem: deploying the Variational Envelope Method, we prove that infinite
meromorphic products of paired zeros and poles blur into smooth, scale-invariant fields
under a sharp density-decay discriminant (Δ𝑛 → 0), and extend this construction to
dynamic trumpet domains.