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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.)

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    You may look for "Shell-Thron-region", which is the complex extension for the "Euler-interval" for real numbers (where the infinite power tower of a real basis converges), see for instance http://math.eretrandre.org/hyperops_wiki/index.php?title=Shell-Thron_region . *i* is inside that region, so the infinite power-tower converges (I think, only the principal branch is always selected). As far as I recall the original article of D. Shell is online available for more details.2012-08-10
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    [Thron's article](http://dx.doi.org/10.1090/S0002-9939-1957-0096054-X) and [Shell's article](http://dx.doi.org/10.1090/S0002-9939-1962-0141910-9). In any event, remember that the power function itself is multivalued...2012-08-10
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    "How do we determine analytically if the tower converges" (http://www.jstor.org/discover/10.2307/2300357) had some analysis for determining the convergence of infinite exponentials. "Both the logarithm and Lambert W functions are multivalued. How do we know which branch to use?" IIRC, the branches of h, are split-level with-respect-to W, i.e. inside the Shell-Thron region you have to use W branch n, and outside the Shell-Thron region you have to use W branch n+-1 (I don't remember), if you want the real part/imaginary part of the result to fall within some specific range.2012-08-17

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