The Lambert transform for small and large values of the transformation parameter

Abstract

We derive asymptotic expansions of the Lambert transform \[ ∫ 0 ∞ x t ( e x t − 1 ) − 1 f ( t ) d t \int _0^\infty xt(e^{xt}-1)^{-1}f(t)dt \] of a locally integrable function f ( t ) f(t) for small and large x x . All the expansions are accompanied by error bounds for the remainder at any order of the approximation.