TITLE: What is the value of $i^{i^{i^\ldots}}$?
QUESTION [5 upvotes]: What is the value of $i^{i^{i^\ldots}}$?

My effort is the following: If $z, \alpha \in \mathbb{C}$ with $z \neq 0$ then we can write $z^{\alpha}=e^{\alpha \log z} = e^{\alpha [ \log |z|+i \text{ Arg z} + 2 \pi i m]}$ where $ m=0, \pm 1, \pm 2, \ldots$  and Arg z is the principle argument of the complex number $z$. But in above expression we don't have the finite no of exponents, so I am not sure how should I use this formula to the solve above problem. Thanks and Regards!

REPLY [5 votes]: Let us denote your power tower by $a$. Then assuming it converges
\begin{gather*}
i^a=a \\
\frac 1i=\left( \frac 1a \right)^{1/a} \\[6pt]
-\ln i =\left( \frac 1 a\right)\ln\left( \frac 1a\right) \\[6pt]
-\ln i=e^{\ln\left( \frac 1a\right)}\ln\left( \frac 1a\right) \\[6pt]
\ln \left( \frac 1a\right) = \mathop{W}(-\ln i) \\[6pt]
\frac 1a=-\frac{\ln i}{\mathop W (-\ln i)} \\[6pt]
\end{gather*}
\begin{align*}
a &= -\frac{\mathop{W}(-\ln i)}{\ln i} \\[6pt]
&= \frac{2i}{\pi}\mathop{W}\left( -\frac 12\pi i\right)\\[6pt]
&\approx 0.44+0.36i
\end{align*}
where $\mathop{W}$ is the Lambert W function.