# Siewert solutions of transcendental equations, generalized Lambert functions and physical applications

## TL;DR

This paper reviews Siewert's exact solutions to transcendental equations, highlighting their expression via generalized Lambert functions and Wright omega functions, with some solutions approximated analytically.

## Contribution

It consolidates and analyzes Siewert's solutions, connecting them with recent developments in generalized Lambert and Wright omega functions, and provides analytical approximations.

## Key findings

- Solutions expressed in terms of generalized Lambert functions.
- Asymptotic forms written as Wright omega functions.
- Analytical approximations for some solutions.

## Abstract

We review the exact solutions of several transcendental equations, obtained by Siewert and his co-workers, in the '70s. Some of them are expressed in terms of the generalized Lambert functions, recently studied by Mez\"o, Baricz and Mugnaini. For some others, precise analytical approximations are obtained. In two cases, the asymptotic form of Siewert's solutions are written as Wright omega functions.