# Julia Size

The central feature of a Julia set associated with a baby Mandelbrot set is a circle.

Let \(f_0 = 0, f_1 = c, f_2 = c^2 + c, \ldots\) be the orbit of the origin.

Consider \(z_0 = \lambda\) on the boundary of the circle.

Then \(z_1 = c + \lambda^2 = f_1 + h\).

Write the Taylor expansion about \(f\):

\(z_p = f_p + \frac{\mathrm{d} f_p}{\mathrm{d} f_1} h + \ldots\).

If \(c\) is periodic with period \(p\), then \(f_p = 0\) and you can pick \(z_0\) on the boundary of the circle such that \(z_0 = z_p\).

This gives \(\lambda = \frac{\mathrm{d} f_p}{\mathrm{d} f_1} \lambda^2\), that is,

\[ \lambda = 1 / \frac{\mathrm{d} f_p}{\mathrm{d} f_1}\]

#include <complex.h>

double _Complex m_julia_size(double _Complex c, int p)
{
    double _Complex z = c;
    double _Complex dz = 1;
    for (int q = 1; q < p; ++q)
    {
        dz = 2 * z * dz;
        z = z * z + c;
    }
    return 1 / dz;
}