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Introduction:

Recently I found out that $i^i \approx 0.20788$ has no imaginary part. I got interested and then wanted to know whether there are other $n$ for which $\underbrace {i^{i^{i^{.^{.^{.^{i}}}}}}}_{n \ times}$ has no inaginary part.

So I wrote this python script (I'm a beginner at python, could be very bad code :) which plots what I call the $i-Tower\ up\ to\ n = 100$. It looks like this: iTower

Question:

Let's use this convention: ${}^ni = \underbrace{i^{i^{i^{.^{.^{.^{i}}}}}}}_{n \ times}$.

  • Why is the sequence ${}^ni\ |\ n \in \mathbb{N}$ converging?

${}^{20}i \approx 0.48770 + 0.41217i$

${}^{60}i \approx 0.437584 + 0.360535i$

${}^{100}i \approx 0.43829+ 0.36059i$

What I already noticed is that the angle between the lines you can draw from ${}^ni$ over ${}^{n+1}i$ to ${}^{n+2}i$ is $<90°$.

  • Is that angle equal for all $n$?

  • Is there a way to give the exact value of $x = {}^ni \ $with$\ {n \rightarrow \infty}$?

  • What is the connection between $i$ and $x$. Is $x$ a special complex number that is maybe already known or comes up in other places?

marked as duplicate by Guy Fsone, Did, Severin Schraven, T. Bongers, Cameron Williams Jan 24 at 19:47

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • I think the same question has been already asked by someone. – Digamma Jan 24 at 13:50
  • 1
    See "Shell-Thron-region" (for area of convergence for complex bases $z$) The hint to Euler and Eisenstein by @JanEerland covers only real $z$ and to generalize this question to the complex numbers has not been trivial... – Gottfried Helms Jan 24 at 16:58
up vote 5 down vote accepted
  • The infinite power tower converges to the value:

$$i^{i^{i^{\cdot^{\cdot^{\cdot^{\cdot}}}}}}=-\frac{\text{W}(-\ln(i))}{\ln(i)}=\frac{2\text{W}\left(-\frac{\pi i}{2}\right)i}{\pi}\approx0.438286+0.3605924i\tag1$$

See: MathWorld

  • Solving this equation:

$$a^b=b\Longleftrightarrow b=-\frac{\text{W}(-\ln(a))}{\ln(a)}\tag2$$

With $\text{W}(z)$ is the product log fuction


Notice, for the 'Product Log Function' (or the Lambert $\text{W}$-function) is defined as follows:

$$f(z)=ze^z\to z=f^{-1}(ze^z)=\text{W}(ze^z)\tag3$$

Now, we know that for $k\in\mathbb{R}^+$ (real, and bigger than zero) $\ln(k)$ is well defined.

Now, your question is about:

$$y(k)=k^{k^{k^{k^{\dots}}}}=-\frac{\text{W}\left(-\ln(k)\right)}{\ln(k)}\tag4$$

Eisenstein's (1844) considered this series of the infinite power tower. $y(k)$ converges iff $e^{-e}\le k\le e^{\frac{1}{e}}$; OEIS A073230 and A073229), as shown by Euler (1783) and Eisenstein (1844) (Le Lionnais 1983; Wells 1986, p. 35).

So, the domain of $y(k)$:

$$\left[0,e^{\frac{1}{e}}\right]\tag5$$

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