# Limit of infinite tetration expression involving the Lambert W function.

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?

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