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.)