The Principal Branch of the Lambert W Function

The Lambert W function is the multi-valued inverse of the function E(z) = z exp z. Let (W) over tilde be a branch of W defined and single-valued on a region (D) over tilde. We show how to use the Taylor expansion of (W) over tilde at a given point of (D) over tilde to obtain an infinite series representation of (W) over tilde throughout (D) over tilde.

Automated summary

The Lambert W function is the multi-valued inverse of a specific function, and this study demonstrates how to use the Taylor expansion of the Lambert W function to obtain an infinite series representation throughout a given region.