Stieltjes, Poisson and similar representations of functions
Corless, Rob, and Kalugin, German
We show that the principal branch of the Lambert function is a Bernstein function. We also show that many functions containing , for example , are Stieltjes functions. The fact that and related functions are Stieltjes, Pick or Bernstein functions leads naturally to integral representations for them, and various integral representations are obtained, including Poisson's integrals. Integral forms based on the Burniston--Siewert method are given as well. Finally, we prove a recent conjecture of B. Jackson, A. Procacci, and A.D. Sokal.
One of the integral representations has been known for some years now, and is on the Lambert~ poster (http://www.orcca.on.ca/LambertW/), but to our knowledge the proof is heretofore unpublished. Several others are new to this work.