Integrals of Infinitely Iterated Functions

Question

As a curiosity, I was looking at functions such as $y = x^{x^{x^{.^{.^{.}}}}}$ and finding their derivatives. I realize this is quite easy. For this problem, we can write $y = x^y$ and use implicit differentiation, and essentially the same concept can be used for any infinitely iterated function like this. I then tried to integrate one of these and am finding it much more difficult. Is anything known about $\int x^{x^{x^{.^{.^{.}}}}}dx$ or any other functions involving some sort of infinite iteration? (like $\sqrt{x + \sqrt{x + ...}}$)?

Answer 1

I would suspect you can show this more rigorously, but here is one way to think about it:

In calculus, finding derivatives of functions with respect to their argument(s), is much easier than computing their indefinite integral. $\int x^xdx$ cannot be expressed in terms of the usual/normal elementary functions we often use. So it usually suffices by taking a more numerical approach, by approximating its value on some defined bound, generating error. Considering the function $f(x)=x$ raised to its argument infinite times, again, analogously to $x^x$, we note its integral yields no closed form expression, and is actually undefined.

As a note, there are various indefinite integrals of functions that don't yield expressions in terms of math's basic elementary functions. As an example, consider $\int \frac {sin(x)}{x}$dx . As it turns out, this integral has no mathematical closed form representation, but we can calculate its definite integral at some intervals.

Answer 2

Indeed, there are closed forms of such integrals. Via user omegadot's answer to the question "Unexpected appearances of $\pi^{2}/6$", we find:

$$I_{1} := \int_{0}^{1} (x^{-x})^{(x^{-x})^{(x^{-x}) \dots }} dx = \frac{\pi^{2}}{6} .$$

We can also express this integral through the Lambert W function - see for instance lemma 3.3 of this paper by Galidakis. As $$f(x):=x^{x^{x^{x^{x^{\dots}} }}} = {^{\infty}x} = - \frac{W_{0}(-\log(x))}{\log(x)} , $$ we obtain $$ I_{1} = \int_{0}^{1}- \frac{W_{0}(-\log(x^{-x}))}{\log(x^{-x})} dx = \frac{\pi^{2}}{6}.$$

Other integrals involving the Lambert W function - which can be rephrased in terms of infinite power towers through the definitions above - can also be obtained.

For instance, on p. 28 of the following paper by Gautschi, we find $$I_{2} := \int_{1}^{\infty} [-W_{0}(-xe^{-x})]^{\alpha} x^{-\alpha} dx = \alpha \psi_{1}(\alpha) -1, $$ where $\psi_{1}(\cdot) $ is the trigamma function and $\alpha>1$. Something similar holds for another branch of the Lambert W function: $$I_{3} := \int_{0}^{1} [-W_{-1}(-xe^{-x})]^{\alpha} x^{-\alpha} dx = \alpha \psi_{1}(1-\alpha)+ 1,$$ where in this case $|\alpha|<1$.

Further integrals involving the Tree T function - which is related to the Lambert W function - can be found in the following paper by Corless, Hu, and Jeffrey.