I have the following julia set: $$ f(z)=z\cdot\Omega=z\cdot\left(\sum_{j=1}^A\omega_j(z)^{-1}\right)^{-1} $$ where $$ \omega_j(z)=C\prod_{i=0}^{N_j}(z-r_{j,i}) $$ Where $C$ and all the $r_{j,i}$ are complex constants. Depending on what I choose for the $r_{j,i}$ I can approximate any shape in the complex plane. Now I would like to estimate the distance to this julia set. I have done this before for simpler fractals using DEM/J, which states that $$ \delta(z)= \lim_{k\to+\infty} \dfrac{|z_k|\log|z_k|}{|z_k'|} $$ Is this approximation for the distance usable with my julia set?
I have tried to derive $f(z)$, but the results I'm getting are not correct distance estimations. I have the following for the derivative $$ (f(z))'=z'\cdot f'(z)=z'\cdot\left(\Omega+z\cdot\Omega^2\cdot\sum_{j=0}^{A}\omega_j'\omega_j^{-2}\right) $$ where $$ \omega_j'(z)=C\sum_{i=0}^{N_j}\left(\prod_{k\neq i}^{N_j}z-r_{j,k}\right) $$ If I understand correctly I should be able to calculate the derivative by doing $z_{k+1}'=f'(z_k)z_k'$.
Is my derivative of $f(z)$ correct?
