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I'm trying to find a book or a paper on infinite exponentiation: more precisely, it should be proving its full interval of convergence on the positive real line, i.e. if $x\in\mathbb{R}^+$, then $x^{x^{x^\cdots}}$ converges if and only if $e^{-e}\le x\le e^{1/e}$.

While I didn't manage to find any books about it, there are these two papers from The American Mathematical Monthly:

  1. Knoebel's Exponentials Reiterated
  2. Mitchelmore's A Matter of Definition

Also, there's this paper on arXiv:

  1. Moroni's The strange properties of the infinite power tower

Every one of them is problematic.

Paper #1 doesn't make sense towards the end of page 242 and is rather incomprehensible there.

Paper #2 shows only the convergence for $1\le x\le e^{1/e}$ and the conclusion is

DEFINITION. For all real $x$ such that $0\le x\le e^{1/e}$, $x^{x^{x^\cdots}}$ is the real solution for $t$ of the equation $x^t=t$, and in case this equation has two solutions, then the lesser one.

That is not satisfactory at all.

Paper #3 makes an incorrect assumption on page 15, which is the following:

If the first first derivative is $|z'(y_1^*)|\lt 1$ (that is $-1\le z'(y_1^*)\lt 0$...

There shouldn't be $-1\le$ but $-1\lt$, the author probably does so to avoid the analysis of the case when $x=e^{-e}$, but at the same time he further in the text states that the infinite tetration for $x=e^{-e}$ is convergent. So that one is incomplete.

1mik1
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    If you only want proofs of the bounds for convergence for infinite tetration, I believe there are proofs in a pair of questions here on mathematics stack exchange. – Mark S. Jan 31 '21 at 13:35
  • @Mark S. I know that, but I want to reference a proof. It will be better if it is a book or a paper. – 1mik1 Jan 31 '21 at 13:37
  • See the comment I feel obliged to do, as a rather experienced man, some days ago to a similar question here – Jean Marie Jan 31 '21 at 14:10
  • @JeanMarie "Tetration has no interest in pure mathematics"? I think it has interest on its own. The research on it gives me headache though... The topic is very obscure. – 1mik1 Jan 31 '21 at 14:46
  • If I were writing a paper for a journal and I needed to cite the convergence interval, I'd try looking for books/papers myself but would be comfortable just citing MSE proofs in the end since this is sort of a folklore result. Can you add the context as to why you need a book or a paper in particular? – Mark S. Jan 31 '21 at 18:35
  • @MarkS. Well, if this is really that folklore result, then books/papers aren't hopefully needed. – 1mik1 Jan 31 '21 at 18:55
  • Euler, Eisenstein should be good search-tokens. I think it was discussed in an exchange between the two people, but don't have a link at hand. – Gottfried Helms Feb 15 '21 at 22:10

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In Ioannis Galidakis' A Collection of References For Infinite Exponentials and Tetration, see items #7, 19, 88, 110, 111, 166. See also On the Convergence of Iterated Exponentiation—I by Creutz/Sternheimer (1980) and On ${x^{x^{x^{{\cdot}^{{\cdot}^{\cdot}}}}}}$ by Louis A. Talman (written in 1999, I think).

Dave L. Renfro
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  • Anderson's paper (item #7) states in the introduction that if $x$ and $y$ are distinct positive numbers such that $x^x=y^y$, then we say the set ${x,y}$ is a Bernoulli pair. This is inconsistent with the rest of the text. I think that Bernoulli pairs should be the solutions of $x^y=y^x$ instead. – 1mik1 Feb 01 '21 at 15:22
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    @1mik1: Yes, I agree. This is clearly a typo, since $x^x = y^y$ for values of $x$ and $y$ greater than $1/e = 0.367879\ldots$ implies $x=y,$ and thus this would not a useful definition. The reason this equality implies $x=y$ is that the function $f:(\frac{1}{e},\infty) \rightarrow {\mathbb R}$ defined by $f(t) = t^t$ is strictly increasing (because its derivative is greater than zero when $\ln t + 1 > 0$), and thus injective, and hence for these values of $x$ and $y$ we have $f(x) = f(y)$ implies $x=y.$ – Dave L. Renfro Feb 01 '21 at 16:37