I'm trying to find a closed form for this integral:$$\int_{0}^{\infty}x^{-x}dx$$
Here's the integrand graph:
Clearly it is convergent. My attempt is to obtain a closed form for the area under the curve. Is this possible?
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3That depends on your notion of closed form. Is $$\sum_{n\geq 1}\frac{1}{n^n}$$ considered as a closed form? – Jack D'Aurizio Jul 13 '17 at 21:52
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2@JackD'Aurizio No, that is most certainly not closed-form. https://en.wikipedia.org/wiki/Closed-form_expression – Franklin Pezzuti Dyer Jul 13 '17 at 21:53
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Then I guess it is not possible, but numerically your integral is extremely close to $2$. – Jack D'Aurizio Jul 13 '17 at 21:55
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2You could consider it a constant. Otherwise $\pi$ would not be closed form as all methods to calculate it are non-finite series expansions. – mathreadler Jul 13 '17 at 21:55
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@mathreadler If one took finitely many elementary functions and their inverses with rational arguments, one could make $\pi$ using trig. – Simply Beautiful Art Jul 13 '17 at 21:58
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1Jack's first answer is probably the nearest you will get ... see https://en.wikipedia.org/wiki/Sophomore%27s_dream – Donald Splutterwit Jul 13 '17 at 21:58
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1@Donald observe that this is a different case because the range of integration is different. – Masacroso Jul 13 '17 at 22:00
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@DonaldSplutterwit: indeed this is not the classical sophomore's dream problem but it boils down to a series similar to $\sum_{n\geq 1}\frac{1}{n^n}$ by splitting the integration range as $(0,1)\cup[1,+\infty)$ and applying simple manipulations. – Jack D'Aurizio Jul 13 '17 at 22:02
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1Thanks for pointing that out ... I had assumed it was just the classic Sophomore's dream ... there is a bit more to it. – Donald Splutterwit Jul 13 '17 at 22:06
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@SimplyBeautifulArt so which functions are considered elementary? Certainly if you consider trig and their inverses elementary you can express $\pi$ on closed form, but then you get away by design. If I consider a function more closely related to the constant I want to express as elementary then I would get away saying it has a closed form. – mathreadler Jul 13 '17 at 22:06
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@mathreadler Generally, elementary functions refers to algebraic functions, exponential functions, trig functions, and their inverses. – Simply Beautiful Art Jul 13 '17 at 22:07
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@SimplyBeautifulArt: Yes I know they are often considered to be, but I don't know why they are natural to consider elementary while so many other non-algebraic functions are not. – mathreadler Jul 13 '17 at 22:12
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@mathreadler And your point here...? – Simply Beautiful Art Jul 13 '17 at 22:14
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1It was more of a curious question than a point. – mathreadler Jul 13 '17 at 22:16
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1We have $$\int_{0}^{+\infty}x^{-x},dx = \sum_{n\geq 1}\frac{1}{n^n}+\int_{0}^{+\infty}\frac{dx}{(1+W(x))e^x}$$ with $W$ being Lambert's function. – Jack D'Aurizio Jul 13 '17 at 22:19
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$e^{-x} = 1/(e^{-x\log(x)})$ – Wolfy Jul 13 '17 at 23:57
3 Answers
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It is highly unlikely that you will be able to find a closed-form expression for your integral. Two special variants of your integral, however, are rather infamous as the "Sophomore's Dream" integrals. They are the integrals $$\int_0^1 x^{x} dx$$ and $$\int_0^1 x^{-x} dx$$ And, as of yet, no closed-form has been obtained for either of them. However, your integral $$\int_0^\infty x^{-x} dx$$ Converges incredibly quickly, even more so than an exponential function, and so a very good approximation can be obtained rather quickly. Wolfram Alpha yields the approximation $$\int_0^\infty x^{-x} dx\approx 1.99545595750014...$$ And so $2$ should be a good enough approximation. Even the inverse symbolic calculator doesn't yield anything for the approximation given by WA, so I doubt that it has any kind of closed form using the elementary functions or any other known constants.
Franklin Pezzuti Dyer
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1One can approximate the remainder to a further extent by using the Euler-Maclaurin formula, which quickly relates the given integral to Sophomore's dream. – Simply Beautiful Art Jul 13 '17 at 22:13
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Not really that strange when you consider how many of these simple-looking integrals it's possible to construct by defining arbitrary operations like tetration. Some of them are bound to be close to integers. As best as I can tell, there's no real physical significance of $x^{-x}$ or of this integral. – Michael L. Jul 13 '17 at 22:31
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We have the following approximation:
Let $S$ be one of the Sophomore's dreams:
$$S=\int_0^1x^{-x}~\mathrm dx=\sum_{n=1}^\infty n^{-n}$$
Then we have the integral $I$ in question,
$$I=\int_0^\infty x^{-x}~\mathrm dx=S+\int_1^\infty x^{-x}~\mathrm dx$$
The second integral has a quick trapezoidal sum approximation:
$$\int_1^\infty x^{-x}~\mathrm dx\approx-\frac12+\sum_{n=1}^\infty n^{-n}$$
Thus, we have
$$I\approx2S-\frac12=2.08257$$
The error is approximately $0.08711$
Simply Beautiful Art
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$\int\limits_0^\infty x^{-x}dx=\int\limits_0^1 x^{-x}dx+\int\limits_0^1 x^{-2+1/x}dx$
The second integral is unpleasant to develope into a series.
But if we can use the solution $\,z_0\,$ for $\,\int\limits_0^1 (x^{xz} - x^{-2+1/x})dx=0$
with $\,z=z_0\approx 1.45354007846425$ , then we get:
$$\displaystyle \int\limits_0^\infty x^{-x}dx=\sum\limits_{n=1}^\infty \frac{1+(-z_0)^{n-1}}{n^n}$$
user90369
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