Abstract
The Lambert W function has a number of integral expressions, including integrals of Bernstein, Thorin, Poisson, Stieltjes, Pick and Burniston–Siewert types. We give explicit integral expressions for W for each of these types. We also give integrals for a number of functions containing W.
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Acknowledgements
We are very grateful to Alan D. Sokal for making interesting suggestions regarding the properties of the Lambert W function. We also thank him for many useful comments and constructive remarks which greatly improved the original manuscript.