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
Solution 1:
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 $$z=\frac{2i}{\pi}\,W(-i\pi /2).$$
Solution 2:
by maple its possible prove this tower is convergence but its numerically proof
