Phase-Resolved Asymptotics for Proximity Functions of Finite Exponential Sums

Abstract

For a finite exponential sum with constant coefficients, we determine the first-order asymptotic of the proximity function at every fixed value different from its constant term. The leading term is the angular measure of the strict decay region multiplied by the negative logarithm of the shifted constant coefficient. The correction has order r^{-1} and is a continuous function of the actual radial phases. Its coefficient is given by real-line logarithmic profiles at the neutral support directions, with the constant plateau subtracted. The expansion holds through all real radii, including circles containing zeros, without a simplicity or zero-separation assumption. A weighted form identifies the corresponding atomic angular correction, and the scaled scalar remainder has a compact interval as its accumulation set. For 1 + e^z we evaluate the coefficient as an explicit piecewise quadratic periodic function, showing that a constant first-order correction need not exist.