Identifying logarithmic tracts

TL;DR

This paper characterizes when direct tracts are logarithmic and provides conditions for their existence, including an example of a function with infinitely many direct singularities but no logarithmic singularity over finite values.

Contribution

It establishes criteria for identifying logarithmic tracts within direct tracts and presents an example of a function with unique singularity properties.

Findings

A direct tract bounded by a simple curve is a logarithmic tract.

Sufficient conditions are given for a direct tract to contain logarithmic tracts.

An example function with infinitely many direct singularities but no finite logarithmic singularities is provided.

Abstract

We show that a direct tract bounded by a simple curve is a logarithmic tract and further give sufficient conditions for a direct tract to contain logarithmic tracts. As an application of these results, an example of a function with infinitely many direct singularities, but no logarithmic singularity over any finite value, is shown to be in the Eremenko-Lyubich class.