# Find the value of $\large i^{i^{.^{.^.}}}$

Find the value of $\large i^{i^{.^{.^.}}}$ ?

How should we start to solve it ?

Also you can see this one if it helps.

Thanks

• Hint: If there is a solution $x$, it has the property $i^x=x$ Jul 18, 2013 at 19:21
• @MahdiKhosravi: Are rast migam. :) Emtehan kon albate khode man ham jozweshun hastama chon untor soalato ghablan ha didam. :D Aug 18, 2013 at 7:27

Let $z=i^{i^{.^{.^{.}}}}$. Then, as Hagen von Eitzen pointed out, $i^{z}=z$. Then $1=z\,i^{-z}=z \, e^{-i\pi z/2}$. It follows that $$-\frac{i\pi}{2}=-\frac{i\pi z}{2}\,e^{-i\pi z/2}.$$ Using the notion of Lambert's W function, we see that $$-\frac{i\pi z}{2}=W(-i\pi/2),$$ or $$z=\frac{2i}{\pi}\,W(-i\pi /2).$$
• Now, is this a solution? That is, does $i,i^i,i^{i^i},\dots$ converge, so $z$ exists? Jul 18, 2013 at 19:31
• Just because $z=x^z$ exists doesn't mean that it makes sense that $x^{x^{x^{\dots}}}=z$. Jul 18, 2013 at 19:32
• Since $i$ is not simply just $\exp\left(i\frac{\pi}{2}\right)$ but also given by $\exp\left(i\frac{\pi}{2}+2k\pi i\right)$, shouldn't there be a family of fixed points? Jul 18, 2013 at 19:32
• @Andres Caicedo: Is it true that $\left|a^{b}\right| \le |a|^{|b|}$? If so, it seems to me that the sequence $y_{0}=i, y_{n+1}=y_{n}^{i}$ is bounded. If bounded, then it must converge. Jul 18, 2013 at 19:45