The Mandelbrot Set and Julia Sets

Change of Variables

For the general, monic polynomial (assuming we've already made the substitution to convert a general polynomial into a monic polynomial)

zn + azn-1 + bzn-2 + ... + cz + d

we substitute

z = w - a/n

and observe

zn + azn-1 + bzn-2 + ... + cz + d = (w - a/n)n + a(w - a/n)n-1 + b(w - a/n)n-2 + ... + c(w - a/n) + d

On the right-hand side, only the first two terms can produce wn-1. How many factors of wn-1 will each term produce?

(w - a/n)n is n factors of (w - a/n) multiplied together. To get wn-1, we need (n-1) copies of w and consequently one copy of -a/n. This happens in n different ways (take -a/n from exactly one of the (w - a/n) factors, w from the rest), so in (w - a/n)n the an-1 term is n(wn-1)(-a/n) = -a(wn-1).

a(w - a/n)n-1 is a multiplied bu n-1 factors of (w - a/n). To get wn-1 we need n-1 copies of w. This happens in exactly one way (take w from all of the (w - a/n) factors), so in a(w - a/n)n-1 the an-1 term is a(wn-1).

So we see that with this substitution, the wn-1 terms cancel.

Return to Change of Variables.