# 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;
}