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For $x \in \Bbb C$, let $f(x)=i^x = \exp(i\pi x)$, where $i^2=-1$. Then find the fixed points for $f$.

EDIT: Let for all $n\geq 1$ $$\large a_n=\underbrace{i^{i^{\cdots i}}}_{\text{$n$ times}}$$

My question is, does the sequence of tetrations $\{a_n\}_{n\geq1}$ converge to some complex number? If yes, then what is it?


marked as duplicate by J. M. is a poor mathematician, Chill2Macht, Willie Wong, Daniel W. Farlow, Claude Leibovici Aug 2 '16 at 3:51

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  • $\begingroup$ How do you define $i^x$ for $x$ in $\mathbb C$? $\endgroup$ – Did Jun 27 '13 at 20:51
  • $\begingroup$ Sorry @Did for not answering before. I define $i^x$ in $\mathbb{C}$ as $i^x=\exp(i\pi (Re(x)+iIm(x))=\exp(-Im(x))e^{(i\pi Re(x)))}$ $\endgroup$ – Samrat Mukhopadhyay Jun 29 '13 at 11:24
  • $\begingroup$ There are several mistakes in the formula you suggest. Once they will be corrected, you could try to solve the system $\Re(i^x)=\Re(x)$, $\Im(i^x)=\Im(x)$ and see what happens. $\endgroup$ – Did Jun 29 '13 at 11:39
  • $\begingroup$ Oops! The edit brings a different question to the fore... Samrat: please do not do that; instead, try to focus on some well-defined question and ask just this question (not several badly defined ones). $\endgroup$ – Did Jun 29 '13 at 11:40
  • $\begingroup$ @Did I had this question in mind that whether infinite tetration can converge to some point. Maybe I have not correctly represented that problem. Could you just put some light what the correct question should be? $\endgroup$ – Samrat Mukhopadhyay Jun 29 '13 at 11:52

We had this question quite recently, as I recall. But I cannot now find it. Anyway, here we see $a_n$ for $n$ from $1$ to $50$. They are converging, right?

enter image description here

The limit is: $$ \frac{2i}{\pi}W\left(\frac{-i\pi}{2}\right) \approx .4382829366+.3605924718 i $$ Of course $W$ is the Lambert W function.


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