TITLE: Complex towers: $i^{i^{i^{...}}}$
QUESTION [16 upvotes]: If $w = z^{z^{z^{...}}}$ converges, we can determine its value by solving $w = z^{w}$, which leads to $w = -W(-\log z))/\log z$.  To be specific here, let's use $u^v = \exp(v \log u)$ for complex $u$ and $v$.
Two questions:

How do we determine analytically if the tower converges?  (I have
seen the interval of convergence for real towers.)
Both the logarithm and Lambert W functions are multivalued.  How do we know which branch to use?

In particular $i^{i^{i^{...}}}$ numerically seems to converge to one value of $i2W(-i\pi/2)/\pi$.  How do we establish this convergence analytically?
(Yes, I have searched the 'net, including the tetration forum.  I haven't been able to locate the answer to this readily.)

REPLY [7 votes]: (Edit-version 2) 
If you have a base $b$ such that you look at $b^{b^{b^{...}}}$ then find a solution for $t$ such that $b = t^{1/t}$ If you have such a $t$, then look at its logarithm $u = \ln(t)$. If $|u| \le 1$ then the infinite tower is a convergent expression. Note that the function h(x), such that $t = h(b) \to t^{1/t} = b $ is multivalued and you take the principal value) I've made a picture about this "Shell-Thron-region" in the tetration-forum (the picture reflects only the upper halfplane, the full picture is symmetric around the x-axis).      
The blue curve indicates the complex bases $b$ on the boundary between convergence and divergence of the resp. infinite powertower. Outside of this curve the powertower diverges. To each point on this curve there is another point $t$ in the complex plane associated (which is on the magenta curve). I connected some example points $b=t^{1/t} $ with a grey line. The yellow curve indicates the points $u$, the logs of the points $t$.     
Inside (and on) the yellow circle (with radius 1 ) are all points $u$ whose exponentials $t$ are inside (and on) the magenta curve and whose associated bases $b$ are inside (and on) the blue curve -and thus whose infinite powertower with base b is convergent.
For values $b$ outside the blue curve, the corresponding $t$ are outside the magenta and the corresponding $u$ are outside the yellow curve the infinite powertower diverges.
(Note, that due to the multivaluedness we can have values $u$ and $t$ outside their curves which have corresponding $b$ inside the blue curve, but that doesn't matter for the question since there must then another value $u$ and $t$ inside their curved regions)
Your example is $b=i$, and that value is inside the blue curve, so the infinite powertower is convergent.