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This weekend I tried to get creative integrals for function compositions of the Lambert $W$ function with elementary functions $W(f(x))$, by using the known as Frullani's theorem, also exploiting formulas that I know from the Wikipedia Lambert $W$ function. In addition I've combined identities (which are natural in the theory of this function), changes of variables, and as a last strategy summation over integers to get the examples below, for the main/principal branch of the Lambert $W$ function denoted in this post as $W(x)$. In this Wikipedia are added the literature justifying the identities that I've used, see please also the corresponding article from the encyclopedia Wolfram MathWorld.

The Wikipedia's page for Frullani integral is this, I know several articles from the literature from my account of JSTOR (say [1] and [2]).

Example 1. We've that the identity $$\int_1^\infty\left(W\left(\frac{1}{W(x^{-1})}+\frac{1}{W(x)}\right)-W(x)\right)\frac{dx}{x}=\frac{\Omega(\Omega+2)}{2}$$ holds, where here and in next examples $\Omega=W(1)$ is the Omega constant.

Example 2. One has the definite integral $$\int_0^\infty \left(W(\Omega x)-W\left(\frac{\Omega x^{\Omega}}{W(x)^{\Omega -1}}\right)\right)\frac{dx}{x^2}=\Omega^2.$$

Example 3. Denoting the Möbius function as $\mu(n)$, then for $\Re s>1$ $$\sum_{n=2}^\infty\frac{\mu(n)}{n^s}\int_0^\infty\frac{W(e^{-x})-W(e^{-nx})}{x}dx=-\Omega\,\frac{\zeta'(s)}{(\zeta(s))^2}$$ where $\zeta(s)$ is the Riemann zeta function (I've created similar expressions than this invoking the dominated convergence theorem).

Question. I would like to know what can be other examples of nice integrals (or identities built from those) arising from the evaluation of certain integrals that involve, in particular, a branch of the Lambert $W$ function (say the same that I've used, $W(x)$, or other branch), and that arise when we invoke the Frullani integral for functions in the integrand related to the Lambert $W$ function*. Many thanks

*I say this to emphasize that you can take function composition or products, changes of variables... of the Lambert $W$ function with other elementary or special functions, and invoke convergence theorems. And for the same or different context, mine was integration over reals.

After there is some answer(s) providing one or several nice examples I should to accept an answer.

References:

[1] A. M. Ostrowski, On Some Generalizations of the Cauchy-Frullani Integral, Proceedings of the National Academy of Sciences of the United States of America Vol. 35, No. 10, pp. 612-616 (1949).

[2] F. G. Tricomi, On the Theorem of Frullani, The American Mathematical Monthly, Vol. 58, No. 3, pp. 158-164 (1951).

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  • $\begingroup$ The motivation is from the importance of this special function in the literature and applications, and the beauty of Frullani's theorem. I believe that isn't in the liteature: integrals for the Lambert $W$ function by using Frullani integral. On the other hand I tried to get different examples for other different special functions than $W(x)$, for example the inverse Gudermannian function (but was failed). $\endgroup$ – user142929 Jan 14 at 14:29
  • $\begingroup$ Many thanks for the upvoter, was generous. As I've evoked then, the rules for an answer is an application of Frullani (or closely related, say Cauchy-Frullani) integral for some function involving the Lambert $W$ function, and you can to use identities from the theory of the Lambert $W$ function and theorems from real or complex analysis (for example convergence theorems) to create your identities/integrals. $\endgroup$ – user142929 Jan 14 at 18:16

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