Limit curves for zeros of sections of exponential integrals

TL;DR

This paper investigates the asymptotic distribution of zeros of partial sums of exponential integral functions, deriving explicit limit curves and analyzing how singularities influence zero behavior, with applications to Bessel functions.

Contribution

It provides explicit limit curves for zeros of exponential integral partial sums and links zero distribution to singularity order, extending understanding of entire function zeros.

Findings

Zeros grow on the order of O(n)

Explicit limit curves for zero distribution are derived

Zero approach rate depends on singularity order

Abstract

We are interested in studying the asymptotic behavior of the zeros of partial sums of power series for a family of entire functions defined by exponential integrals. The zeros grow on the order of O(n), and after rescaling we explicitly calculate their limit curve. We find that the rate that the zeros approach the curve depends on the order of the singularities/zeros of the integrand in the exponential integrals. As an application of our findings we derive results concerning the zeros of partial sums of power series for Bessel functions of the first kind.