The Mandelbrot Set and Julia Sets

Change of Variables

We have seen we may assume the general cubic polynomial has this form

z3 + az2 + bz + c

Motivated by the quadratic case, we try the substitution

z = w - (a/3)

Then

z3 + az2 + bz + c = (w - (a/3))3 + a(w - (a/3))2 + b(w - (a/3)) + c

After some simplification this becomes

w3 + (b - a2/2)w + (c - ab/3 + 2a3/27)

Return to Change of Variables.