Series Expansions of Lambert W and Related Functions
Abstract
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' 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.