Some definite integrals containing the Tree T function

Abstract

We consider, from a symbolic point of view, a pair of definite integrals containing Lambert W , recently considered from a numerical point of view by Walter Gautschi. We transform the integrals to a shape that can be integrated in special cases by a computer-algebra system or by using tables of integrals, such as Prudnikov et al.

Errata and Addenda to Mathematical Constants

“…Also, 3 −1 e −1/3 = 0.2388437701... arises in [892] as a consequence of the formula −W (−3 −1 e −1/3 ) = 1/3. Note that −W (−x) is the exponential generating function for rooted labeled trees and hence is often called the tree function [893].…”