I notice both wikipedia and mathworld have the great result of $\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx$ that:

$\int\underbrace{x^{x^{\cdot^{\cdot^x}}}}_m~dx=\sum\limits_{n=0}^m\dfrac{(-1)^n(n+1)^{n-1}}{n!}\Gamma(n+1,-\ln x)+\sum\limits_{n=m+1}^\infty(-1)^na_{mn}\Gamma(n+1,-\ln x)$ , where $a_{mn}=\begin{cases}1&\text{if}~n=0\\\dfrac{1}{n!}&\text{if}~m=1\\\dfrac{1}{n}\sum\limits_{j=1}^nja_{m,n-j}a_{m-1,j-1}&\text{otherwise} \end{cases}$

How does this result derived?

  • 5
    $\begingroup$ With all due respect for tetration (very little, as far as I am concerned), this is one truly ugly result, imfho. I really hope they can find some good application of this, either in some applied or pure mathematics field, in some "real life" one or whatever, otherwise it is going to remain just an ugly result. The question though is interesting: how did they get it? +1 $\endgroup$
    – DonAntonio
    Apr 6, 2014 at 11:19
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    $\begingroup$ I think this result is beautiful, not ugly. And I couldn't care less if it doesn't have any "applications". $\endgroup$
    – user85798
    May 19, 2014 at 23:58

1 Answer 1


Somebody posted a nice question on sci.math long time ago about finding the expansion of the function exp(exp(x)) in terms of x. Two people answered correctly: Robert Israel and Leroy Quet. I used Leroy's result and found that a secondary pattern emerged, which is the series up to m, which is related to the W function, for which I later published (this and some other results as incidental by indcution). In reality Leroy's name should be on the reference, since he solved the basic step of finding the first expansion. At the time I underestimated the importance of his result and forgot to give him credit for it.

  • $\begingroup$ Could you provide a link to the paper? $\endgroup$ May 20, 2014 at 0:10
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    $\begingroup$ @Antonio Vargas: link $\endgroup$
    – user127032
    May 20, 2014 at 0:16

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