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We can define ${}^nx$ as $\underbrace{\displaystyle {x^{x^{\cdot ^{\cdot ^{x}}}}}}_{n\text{ times}}$ (Tetration). I conjecture that ${}^ni$ is complex for all $n \ge 3, n \in \mathbb{N}$.

I've attempted to prove this via induction and the exponential form of of a complex number ($i = e^{i\pi/2}$), and believe that I succeeded. However, my proof is rather shaky, and I'm wondering about a more rigorous proof? So, how would you prove

${}^ni \in \mathbb{C}, \forall n \ge 3, n \in \mathbb{N}$


I won't include my full proof as it's rather long, but a quick summary is something like this:

We'll use proof by induction, starting with $n = 3$

Let $n = 3$. Therefore

$$ \begin{align} {}^3i & = i^{i^i} \\ & = i^{e^{-\pi/2}} \\ & = (e^{i\pi/2})^{e^{-\pi/2}} \\ & = e^{(i\pi e^{-\pi/2})/2} \\ & = \cos(\frac{\pi}{2} e^{-\pi/2}) + i\sin(\frac{\pi}{2} e^{-\pi/2}) \\ & \in \mathbb{C} \end{align} $$

Now, assume that the statement is true for $n = k$, i.e.

$$ {}^ki = \cos\theta + i\sin\theta, \: \theta \ne m\pi $$

Let $n = k+1$. Therefore

$$ \begin{align} {}^ni & = {}^{k+1}i \\ & = i^{({}^ki)} \\ & = i^{\cos\theta + i\sin\theta} \\ & = i^{\cos\theta}i^{i\sin\theta} \\ & = (e^{i\pi/2})^{\cos\theta}(i^i)^{\sin\theta} \\ & = (e^{(i\pi\cos\theta)/2})(e^{-\pi/2})^{\sin\theta} \\ & = r\left(\cos\left(\frac{\pi}{2}\cos\theta\right) + i\sin\left(\frac{\pi}{2}\cos\theta\right)\right) \\ & \in \mathbb{C} \end{align} $$

where $r = e^{-\pi/2}$. Now, I thought that the only way this may not be complex is if $\sin\left(\frac{\pi}{2}\cos\theta\right) \ne 0$, so I performed proof by induction a second time on $\theta$, which proved that $\sin\left(\frac{\pi}{2}\cos\theta\right) \ne 0$ for the relevant $\theta$.

In addition, I've verified it using this Jelly program up to $n = 1000$


As a side note, is this a "known" fact/proof?

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  • $\begingroup$ By "complex" you mean "not real" ? $\endgroup$ – Dominique Mattei Oct 17 at 15:00
  • $\begingroup$ @DominiqueMattei Yes i.e. non-zero imaginary part $\endgroup$ – caird coinheringaahing Oct 17 at 15:00
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    $\begingroup$ I just want to note that there are infinitely many possible definitions of $\text{i}^\text{i}$, for example. It is perfectly reasonable to define $\text{i}^\text{i}$ to be $$\text{e}^{-\left(2n+\frac14\right)\,\pi}$$ for any integer $n$. $\endgroup$ – Batominovski Oct 17 at 15:01
  • $\begingroup$ @Batominovski I can clarify to mean the principle value for each ${}^ni$ in the question if necessary $\endgroup$ – caird coinheringaahing Oct 17 at 15:01
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    $\begingroup$ You cannot assume ${^k}\text{i}=\cos(\theta)+\text{i}\,\sin(\theta)$, though. There is no reason to believe that ${^k}\text{i}$ has unit modulus. $\endgroup$ – Batominovski Oct 17 at 15:07
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At least, the iterations with $n$ increasing walk to a complex fixpoint which is so distant to the real axis that only for a finite number of iterations the orbit crosses the real axis (and might possibly have a "real only" value) and after that all iterations spiral around the fixpoint $t^+$ (with positive imaginary part) towards that fixpoint.

Let $z=1$ and $z_n$ or $\;^nz$ be the $n$'th tetration of the complex unit $I$ . We have then:

 n   z_n                                 imaginary part
--------------------------------------------------------
 0   1                               --> 0
 1                  1*I              --> 1
 2 0.207879576351 + 0*I              --> 0.0
 3 0.947158998072 + 0.320764449979*I --> 0.320764449979
 4 0.050092236109 + 0.602116527036*I --> 0.602116527036
 5 0.387166181086 + 0.030527081605*I --> 0.0305270816055
 6 0.782275682434 + 0.544606557658*I --> 0.544606557658
 7 0.142561823164 + 0.400466525337*I --> 0.400466525337
 8 0.519786396408 + 0.118384196416*I --> 0.118384196416
 9 0.568588617272 + 0.605078406798*I --> 0.605078406798
10 0.242365246825 + 0.301150592071*I --> 0.301150592071
11 0.578488683377 + 0.231529735307*I --> 0.231529735307
12 0.427340132692 + 0.548231217344*I --> 0.548231217344
13 0.330967104358 + 0.262891842795*I --> 0.262891842795
14 0.574271015390 + 0.328715623630*I --> 0.328715623630
15 0.369948042157 + 0.468173193372*I --> 0.468173193372
16 0.400633494867 + 0.263120213351*I --> 0.263120213351
...   ...                                  ...
-------------------------------------------------------
 approximating the fixpoint t^+
   0.438282936727 + 0.360592471871*I    

One can define the iterates $ \;^nz$ as log-polar values of $\xi_n= \; ^nz - t^+$ such that we write $\xi_n= \{ \lambda_n, \varphi_n \} $ .

It is known that the distance to the fixpoint decreases continuously ($ \lambda_n$ is the log of the distance at $n$'th iterate) being interpolateable by the Schroeder-fractional iterates to a smooth almost spiral flow which converges away from the real axis.

image1

image2

At the end, the picture using Schroeder-mechanism for interpolation to fractional iterates, giving a smooth curve. The unit-intervals for one iteration are marked by different colors.
It is proven that this curve is in coincidence with the natural iterates $\{0,1,z,\;^2z,...\}$ and moreover that it is smooth, so this proves the impossibility of $\;^n i$ being real for $n>3$.

image3

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