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The infinite tetration ${\displaystyle z^{z^{z^{\cdot ^{\cdot ^{\cdot }}}}}\!}$ extended to the complex plane is expressed in terms of the Lambert W function as

${\displaystyle {\frac {W(-\ln(z))}{-\ln(z)}}}$

It is possible to verify with the Mathematica CAS that the limit of the expression above as $x$ approaches zero is equal to zero. The result is also verifiable numerically.

Is it possible to find, prove or verify the value of this limit analytically?

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  • $\begingroup$ It is simple if $z$ is a real. $\endgroup$ – Claude Leibovici Apr 27 at 4:13

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