Robert P. Munafo, 2003 Sep 26.
The derivative of the Mandelbrot iteration function is taken with respect to C, and is computed as follows:
Z0 = 0
d/dCZ0 = 0
Z1 = Z02 + C = C
d/dCZ1 = 2 Z0 d/dCZ0 + 1 = 1
Z2 = Z12 + C = C2 + C
d/dCZ2 = 2 Z1 d/dCZ1 + 1 = 2 Z1 + 1 = 2 C + 1
Z3 = Z22 + C = C4 + 2 C3 + C2 + C
d/dCZ3 = 2 Z2 d/dCZ2 + 1 = 2 Z1 + 1 = 4 C3 + 6 C2 + 2 C + 1
Z4 = Z32 + C
d/dCZ4 = 2 Z3 d/dCZ3 + 1
The derivative has many applications:
It is the main part of the formula for the distance estimator, the best way to create images of the Mandelbrot set.
Using Newton's Method, the derivative can be used to locate all the points of bifurcation (bond points) from a mu-atom to its children. Together with the mu-atom size formulas, this can be used to locate all of the descendants of any given mu-atom.
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013. Mu-ency index