Asymptotic values of entire functions of infinite order

TL;DR

This paper constructs an entire function of infinite order with arbitrarily slow growth that attains every complex number as an asymptotic value, demonstrating the diversity of asymptotic behaviors possible.

Contribution

It proves the existence of an entire function of infinite order with universal asymptotic values and slow growth, expanding understanding of complex function behavior.

Findings

Existence of entire functions with all complex numbers as asymptotic values

Construction of functions with arbitrarily slow growth of infinite order

Demonstration that infinite order allows universal asymptotic value behavior

Abstract

We prove that there exists an entire function for which every complex number is an asymptotic value and whose growth is arbitrarily slow subject only to the necessary condition that the function is of infinite order.