The asymptotic zero-counting measure of iterated derivaties of a class of meromorphic functions

TL;DR

This paper derives an explicit formula for the asymptotic distribution of zeros of high-order derivatives of a class of meromorphic functions, extending Pólya's Shire theorem with measure-theoretic insights.

Contribution

It provides a new explicit formula for the zero-counting measure of derivatives of functions combining rational and polynomial parts, extending existing theorems.

Findings

Explicit formula for zero-counting measure derived

Extension of Pólya's Shire theorem to broader class of functions

Enhanced understanding of zero distribution in complex analysis

Abstract

We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence $\left\{\frac{\mathrm{d}^n}{\mathrm{d}z^n}\left(R(z)\exp{T(z)}\right)\right\}$. Here, $R(z)$ is a rational function with at least two poles, all of which are distinct, and $T(z)$ is a polynomial. This is an extension of a recent measure-theoretic refinement of P\'olya's Shire theorem for rational functions.