Reciprocal
We have seen that the inversion considered as a complex function corresponds to a geometric transformation which is pretty easy to define. But it has a conjugation in its formula. Now let us examine below the reciprocal function which is a bit more simple in this regard.
\(f(z) = 1/z = \overline{1/\overline{z}},\quad z \in \mathbb{C}(0,3)\)
Here also we experience the swap of light and dark values. Although the angle of each color is not fixed anymore but is reflected on the real axis. This phenomena becomes obvious after one transformation—see the formula above—so the reciprocal function is the conjugate of the inversion.
The reciprocal function is not defined at zero: it is a singularity, in fact a pole of order one. One could recognise it being a pole through the concentration of white, and its order can be read similar to the method described in the case of zeros. (Going round once the colors repeat one time.)