The Mandelbrot Set and Julia Sets

Change of Variables

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

z2 + az + b

We eliminate the az term by completing the square:

z2 + az + b = z2 + az + (a/2)2 - (a/2)2 + b

= (z + (a/2))2 + (b - (a/2)2)

So the substitution w = z + (a/2) gives

z2 + az + b = w2 + (b - (a/2)2)

For later use we reformulate the substitution: z = w - (a/2). Then

z2 + az + b = (w - (a/2))2 + a(w - (a/2)) + b = w2 + (b - (a/2)2)

Return to Change of Variables.