## Thesis Format

Integrated Article

# Series Expansions of Lambert W and Related Functions

## Degree

Master of Science

## Program

Applied Mathematics

## Supervisor

Jeffrey, David J.

## 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.

## Summary for Lay Audience

Back in secondary school, one may have learned about functions in math class. One property that is fundamental

to the definition of a function is the so-called "vertical-line test": a curve represents a function if a vertical line passes

through said curve only once. In the same class, one may remember learning about inverses of functions, which "cancel"

out the other function and leaves only the input. For example, the function $f^{-1}$ is the inverse of another function

$f$ if $f^{-1}(f(x))=f(f^{-1}(x))=x$.

When combining these two ideas, some contradictory examples may arise. There has been another oft-forgotten aspect

of math not yet mentioned: the horizontal line test. When a horizontal line passes through a curve only once, the corresponding

function has an inverse. However, higher-level mathematics ignores this rule for some cases. The Lambert $W$ function is one

such case, being the inverse of a function which fails the horizontal line test.

As we must still obey the vertical line test, sections of the Lambert $W$ function are treated as separate entities called "branches".

Due to $W$ being a function which is impossible to write out in simpler terms, we need to represent each branch in another way. This

thesis focuses on the use of sums of other functions to approximate the branches of $W$. In addition, we use the same sums

to approximate branches of functions similar to $W$ to lay groundwork for future functions which have branches.

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