Complex Generalisation of Infinite Tetration

Using substitutions and rearrangements, we are going to write $\displaystyle y= ^{\infty} z$ in the form of $\displaystyle x=W(x)e^{W(x)}$ (So we are looking after $y$ ).

Since we have an infinite tetration : $\displaystyle y= ^{\infty} z = z^{z^{z^{.^{.^{z}}}}} = z^y$

$\displaystyle y=z^y = e^{y\ln z} <=> ye^{-y\ln z} =1$

Multiply both sides by $-\ln z$ :

$\displaystyle (-y\ln z)e^{-y\ln z} = -\ln z$

now define $x= -\ln z$ and $f(x) = -y\ln z$

$\displaystyle => x=f(x)e^{f(x)}$ therefore $f(x) = W(x)$

now substitute back $x$ and $f(x)$ :

$\displaystyle -y\ln z = W(-\ln z) =>\boxed{ y= \frac{W(-\ln z)}{-\ln z}}$