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We show that many functions containing $W$ are Stieltjes functions. ExplicitStieltjes integrals are given for functions $1/W(z)$, $W(z)/z$, and others. Wealso prove a generalization of a conjecture of Jackson, Procacci & Sokal.Integral representations of $W$ and related functions are also given which areassociated with the properties of their being Pick or Bernstein functions.Representations based on Poisson and Burniston--Siewert integrals are given aswell.

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