Integral with Lambert $W$ function

Asked Nov 06 '22 at 15:03

Active Nov 07 '22 at 02:39

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I'm trying to calculate the integral of the infinite tetration of $x$ where it's defined,

$$\displaystyle\int_{e^{-e}}^{e^{1/e}}x^{x^{x^{x^{x^{...}}}}}dx$$

which simplifies to $\displaystyle\int_{e^{-e}}^{e^{1/e}}\frac{W(-\ln x)}{-\ln x}\,dx$ or $\displaystyle\int_{-1/e}^{e}\frac{W(u)}{u}e^{-u}\,du$. However, I don't know where to go from there. I'm sure there isn't an elementary antiderivative but WolframAlpha says it's about $1.244$.

edited Nov 07 '22 at 02:39

metamorphy

39,111

asked Nov 06 '22 at 15:03

Noa Arvidsson

This doesn't have a full answer, but does this help? – Accelerator Nov 07 '22 at 01:56
1

$\displaystyle=\int_{-1}^1(1+x)e^{-xe^x},dx.$ – metamorphy Nov 07 '22 at 02:38
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Sorry, but does this answer your question? Integral over domain of infinite tetration of x over extended domain from 0 to $\sqrt[e]e$. Possible $\int_{e^{-e}}^{e^\frac1e} x^{x^{…}}dx$ solution. – Тyma Gaidash Nov 07 '22 at 03:17

Integral with Lambert $W$ function

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