5

Is the following improper integral known? $$\int_{0}^{\infty} x^{-x} \, dx$$ Wolfram Alpha gives both the proper integral from 0 to 10 and 0 to 10000 as 1.99546 so I'm assuming it's equal to 2 (though it could actually be 1.99546), but is this known?

Michael Hardy
  • 1
pommicket
  • 151
  • I sincerely doubt that $2$ is the exact value. I'm too lazy to prove it at the moment, though. – MPW Jul 31 '16 at 03:49
  • Maybe this is remote from anything you care about, but in one sense this is not an improper integral. Since $\displaystyle\int_0^\infty \left| \frac{\sin x} x \right|, dx=+\infty,$ you can't define $\displaystyle\int_0^\infty \frac{\sin x} x, dx$ except as a limit $\displaystyle \lim_{a\to\infty} \int_0^a \frac{\sin x} x,dx$. But since this one does not have positive and negative parts separately diverging to infinity, Lebesgue's definition of the integral applies on the interval $(0,\infty)$ in just the way it does on a bounded interval, so no limit as $a\to\infty$ is needed. $\qquad$ – Michael Hardy Jul 31 '16 at 03:53
  • We can say $\int_{10000}^\infty x^{-x}dx \lt \int_{10000}^\infty x^{-10000}dx=-\frac 1{9999}x^{-9999}|_{10000}^\infty \approx 10^{-4}(10^4)^{10^4}=10^{4\cdot 10^4+4}$ so those digits should be right and the integral is not $2$ – Ross Millikan Jul 31 '16 at 03:54
  • 1
    The difference $2-1.99546$ seems too big to be $\displaystyle \int_{10000}^\infty x^{-x},dx. \qquad$ – Michael Hardy Jul 31 '16 at 03:56
  • I thought 1.99546 could be a rounding error or something like that. – pommicket Jul 31 '16 at 03:56
  • 1
    @MichaelHardy I agree, there's unlikely any sort of machine precision error causing 1.99546, but oh man is it tantalizingly near 2. – Carser Jul 31 '16 at 03:57
  • 1
    The value does not have anything to do with $2$. The numerical values of the integral up to $\infty$ is $1.9954559575001380005$ and inverse symbolic calculators do not find anything (which is normal). Somophore's dream. – Claude Leibovici Jul 31 '16 at 04:42
  • 1
    For information : https://fr.scribd.com/doc/34977341/Sophomore-s-Dream-Function – JJacquelin Jul 31 '16 at 06:50

0 Answers0