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:
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) 1987-2023.
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2023 Mar 31. s.27