Physical Sciences and Mathematics Commons^™

Series Expansions Of Lambert W And Related Functions, Jacob Imre

Electronic Thesis and Dissertation Repository

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 …