Stieltjes, Poisson and other integral representations for functions of Lambert $W$

German A. Kalugin, David J. Jeffrey, Robert M. Corless

We show that many functions containing $W$ are Stieltjes functions. Explicit Stieltjes integrals are given for functions $1/W(z)$, $W(z)/z$, and others. We also prove a generalizat…