Series Expansions of Lambert W and Related Functions

In the realm of multivalued functions, certain specimens run the risk of being elementary or complex to a fault. The Lambert $W$ function serves as a middle ground in a way, being non-representable by elementary functions yet admitting several properties which have allowed for copious research. $W$ utilizes the inverse of the elementary function $xe^x$, resulting in a multivalued function with non-elementary connections between its branches. $W_k(z)$, the solution to the equation $z=W_k(z)e^{W_k(z)}$ for a branch number $k \in \Z$, has both asymptotic and Taylor series for its various branches. In recent years, significant effort has been dedicated to exploring the further generalization of these series. The first section of this thesis focuses on the generalization and representation of series for any branch of the Lambert $W$ function. Rather than the principal branch in the real plane, non-principal branches are of most interest. Behaviour of these branches\u27 approximations is studied near branch cuts and for large-indexed branches. This analysis is supported by both images of curves in domain space and domain-colouring of entire regions. Subsequent sections will focus on a new class of functions which resemble Lambert $W$. These share a fundamental relation with the Lambert $W$, enabling the previous series to be generalized even further. The complexity of the nested functions will increase throughout these sections. Initially, functions are utilized that are multivalued in a single, elementary fashion. Later, these will be replaced with functions which have more complex branch behaviour

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

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.Comment: 17 pages, 5 figure

A New Effective Asymptotic Formula for the Stieltjes Constants

We derive a new integral formula for the Stieltjes constants. The new formula permits easy computations as well as an exact approximate asymptotic formula. Both the sign oscillations and the leading order of growth are provided. The formula can also be easily extended to generalized Euler constants

On the chromatic roots of generalized theta graphs

The generalized theta graph \Theta_{s_1,...,s_k} consists of a pair of endvertices joined by k internally disjoint paths of lengths s_1,...,s_k \ge 1. We prove that the roots of the chromatic polynomial $pi(\Theta_{s_1,...,s_k},z) of a k-ary generalized theta graph all lie in the disc |z-1| \le [1 + o(1)] k/\log k, uniformly in the path lengths s_i. Moreover, we prove that \Theta_{2,...,2} \simeq K_{2,k} indeed has a chromatic root of modulus [1 + o(1)] k/\log k. Finally, for k \le 8 we prove that the generalized theta graph with a chromatic root that maximizes |z-1| is the one with all path lengths equal to 2; we conjecture that this holds for all k.Comment: LaTex2e, 25 pages including 2 figure