The complex function $\log(z) / \sqrt{z}$ is a curiosity that I find interesting since one can express $e^{i\pi}+1=0$ as $\log(-1) / \sqrt{-1} = \pi$.
My question is, what is the significance of the following complex values?
$\log(5.06982105...+i\times2.16077849...) / \sqrt{5.06982105...+i\times2.16077849...} = $ $\qquad0.746670201...+i\times0.0226809289...$
$\log(0.746670201...+i\times0.0226809289...) / \sqrt{0.746670201...+i\times0.0226809289...} =$ $\qquad -0.336892113...+i\times0.0402541188...$
$\log(-0.336892113...+i\times0.0402541188...) / \sqrt{-0.336892113...+i\times0.0402541188...} =$ $\qquad 5.06982105...+i\times2.16077849...$
It appears that the repeated iteration of this function almost always eventually leads to this cycle of three values.
I was expecting these values to be somehow related to the Lambert $W$ function, but I cannot even find such a connection, nor any other connection to known constants, functions, or values.
Also, why would this cycle have three values rather than two values?
For the record, there is one complex number that satisfies $\log(z) / \sqrt{z} = z$:
$e^{(-2/3)W(-3/2)}$
But when one begins with randomly chosen complex numbers, the iteration of this function leads not to this value, but to the cycle of three values given above.
Based on John Barber's comment, I plotted the fractal he mentioned using the domain coloring below:
The above image assigns a color to every point in the complex plane (only $[-10,10]$ of each axis is shown). Plotting $\log(z)/\sqrt z$ gives:
And iterating 100 times gives:
There appears to be 5 colors: yellow, dark yellow, brown, dark brown, and red (two shades, probably). This suggests that after infinite iterations, there can only be those few values. From the original domain coloring, it's clear that yellow and dark brown are complex conjugates, dark yellow and brown are conjugates, and the two reds are conjugates.
Someone with more skill, time, and patience can probably explain the fractal domain boundaries.