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I don't know if this integral was in the literature, then refers it please. This question is a curiosity after I did some experiments with Wolfram Alpha online calculator, and I hope that has mathematical meaning, that is, that I believe that it is well-defined.

Consider the sequence of integrals with first term
$$\int_1^{\infty}x^{-x}dx,$$ and second $$\int_1^{\infty}\int_1^{\infty}x^{-y}y^{-x}dxdy.$$

And now the third term of this sequence being a triple integral, with factors in this new integrand $x^{-y}, y^{-x}$ and $z^{-x}, z^{-y}$, and also $z^{-x}$ and $z^{-y}$.

Thus I am saying this integral $$\int_1^{\infty}\int_1^{\infty}\int_1^{\infty}x^{-y-z}y^{-x-z}z^{-x-y}dxdydz.$$ And the next integral of our infinite sequence of integrals, with a similar pattern than previous.

Question. Is it known what is the value of the limit integral? If you know the literature please refers it. If it isn't in the literature, can you then calculate an approximation of such limit integral? Or well a good upper bound. Many thanks.

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    Oh dear, save me from this madness... – Simply Beautiful Art Jul 18 '17 at 19:13
  • The first integral is discussed here https://math.stackexchange.com/questions/2358024/integral-int-0-inftyx-xdx/2358048#2358048 ... it is not Sophomore's dream ! – Donald Splutterwit Jul 18 '17 at 19:14
  • Wait, you are asking for the limit of these integrals? – Simply Beautiful Art Jul 18 '17 at 19:14
  • Yes I know @DonaldSplutterwit many thanks. –  Jul 18 '17 at 19:18
  • Many thanks @SimplyBeautifulArt It is a curiosity from the ignorance. Any case I am waiting what are saying the users, and I accept your words. –  Jul 18 '17 at 19:19
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    It is a good exercice for you. So instead of waiting other people work for you, what did you try ? – reuns Jul 18 '17 at 19:59
  • Your are right. My apologizes to you @reuns and the other users. Currently I do questions in this site, and maybe it is as a subterfuge to don't study from books. I don't know how work with mathematics, from a serious way. Thus I don't know how solve this exercise. –  Jul 18 '17 at 20:11

1 Answers1

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Conjecture: they tend to zero:

$\int_1^{\infty } x^{-x} \, dx\approx 0.70417$

$\int _1^{\infty }\int _1^{\infty }x^{-y} y^{-x}dydx\approx 1.2259$

$\int _1^{\infty }\int _1^{\infty }\int _1^{\infty }x^{-y-z} y^{-x-z} z^{-x-y}dzdydx\approx 0.120842$

$\int _1^{\infty }\int _1^{\infty }\int _1^{\infty }\int _1^{\infty }x^{-w-y-z} y^{-w-x-z} z^{-w-x-y} w^{-x-y-z}dwdzdydx\approx 0.0116407$

Raffaele
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