If the convergence is sufficiently uniform (so the derivatives converge), we have |
(d/dε)fn(-2 + ε/rn)|ε=0 = gn'(ε)|ε=0 = g'(ε)|ε=0 |
By the chain rule, the equation becomes |
((d/dc)fn(c)|c=-2)⋅((d/dε)(-2 + ε/rn)|e=0) = g'(0). |
Motivated by the slope of f3 at the 3-cycle midget cardioid center, we take |
g'(0) = -1. |
With this, we find |
rn = -fn'(-2) |
The relation |
fn+1(c) = (fn(c))2 + c |
implies |
fn+1'(c) = 2⋅fn(c)⋅fn'(c) + 1. |
Using |
fn'(-2) = -4n/6 - 1/3, |
for n > 1. Dropping the -1/3, small compared to the other term for large n, we take the scaling factor to be |
rn = 4n/6. |
Return to Hurwitz-Robucci scaling.