$\int_{0}^{x} t^tdt$

Asked Sep 11 '17 at 11:35

Active Sep 11 '17 at 11:40

Viewed 97 times

Is there a closed analytical expression for $I(x)$ with $x,t \in \mathbb{R_+}$ and

$$I(x) = \int_{0}^{x} t^tdt$$ ?

I suppose this integral is some classic example for a function without analytic closed form. I have searched this site the web and testet several CA systems but could not find a result.

Question 2: In case there is not, is there way to show this?

edited Sep 11 '17 at 11:40

asked Sep 11 '17 at 11:35

Raphael J.F. Berger

1,697
14
25

3

This may be relevant. – Zubzub Sep 11 '17 at 11:46
$t^t=e^{t\ln t}$ – Olba12 Sep 11 '17 at 11:46
@Olba12: thats entirely clear to me, I just did not see before Zuzub's comment how this contributes to the solution. – Raphael J.F. Berger Sep 11 '17 at 11:48
2

Please see this https://math.stackexchange.com/questions/141347/finding-int-xxdx – Olba12 Sep 11 '17 at 11:49
1

https://fr.scribd.com/doc/34977341/Sophomore-s-Dream-Function – JJacquelin Sep 11 '17 at 13:09
@JJacquelin: So its $Sphd(1;x)$ ! – Raphael J.F. Berger Sep 11 '17 at 13:47
Yes, it is. Insofar this special function becomes referenced as a standard function. – JJacquelin Sep 11 '17 at 14:15

$\int_{0}^{x} t^tdt$

0 Answers0