The Lambert W-functions and some of their integrals: a case study of high-precision computation

Abstract

The real-valued Lambert W-functions considered here are w ₀(y) and w _− 1(y), solutions of we ^w = y, − 1/e < y < 0, with values respectively in ( − 1,0) and ( − ∞ , − 1). A study is made of the numerical evaluation to high precision of these functions and of the integrals $\int_1^\infty [-w_0(-xe^{-x})]^\alpha x^{-\beta}\d x$, α > 0, β ∈ ℝ, and $\int_0^1 [-w_{-1}(-x e^{-x})]^\alpha x^{-\beta}\d x$, α > − 1, β < 1. For the latter we use known integral representations and their evaluation by nonstandard Gaussian quadrature, if α ≠ β, and explicit formulae involving the trigamma function, if α = β.