Remarks on the tangent function from an analytic and probability point of view

TL;DR

This paper explores the properties of the tangent function from an analytic and probabilistic perspective, establishing its connection to Pick functions and classical infinitely divisible measures.

Contribution

It demonstrates that tan(1/it) is a Pick function and links it to probability, also identifying its classical counterpart via Rademacher series.

Findings

tan(1/it) is a Pick function and a free-infinitely divisible transform

Established connections between the tangent function and probability theory

Identified a classical infinitely divisible measure related to the tangent function

Abstract

We show that a function $\tan(1/it)$ is a Pick function (free-infinitely divisible transform) and indicate its connections with a probability. Moreover, we found its "counterpart" in classical infinitely divisible measures expressed as series of Rademacher variables.