# Prove that ${}^ni$ is complex for all $n \ge 3$

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?

• By "complex" you mean "not real" ? – Dominique Mattei Oct 17 at 15:00
• @DominiqueMattei Yes i.e. non-zero imaginary part – caird coinheringaahing Oct 17 at 15:00
• 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 $n$. – Batominovski Oct 17 at 15:01
• @Batominovski I can clarify to mean the principle value for each ${}^ni$ in the question if necessary – caird coinheringaahing Oct 17 at 15:01
• 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. – Batominovski Oct 17 at 15:07

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.

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 $$n>3$$.