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

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

2011-03-27

arxiv.org/abs/1103.5640v1

Mathematics

Complex Variables

Scientific paper

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 generalization of a conjecture of Jackson, Procacci & Sokal. Integral representations of $W$ and related functions are also given which are associated with the properties of their being Pick or Bernstein functions. Representations based on Poisson and Burniston--Siewert integrals are given as well.

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