Mandelbrot iteration
Let us define another function series according to the these formulas: \[M_0(z) = z,\quad M_{n+1}(z) = M_{n}(z)^2 + z.\quad (n = 0, 1, \dots)\]
One may recognise the iteration which is used to define the Mandelbrot set. This is maybe the most widely known fractal. Traditionally we examine the iteration starting from each point, and color the point according to some properties of the resulting sequence. Now we act as if doing the steps of the iteration simultaneously on the whole complex plane.
\(f(z) = M_{n}(z),\quad z \in \mathbb{C}(-\frac{1}{2},2)\)
It may sound weird but now we have approximated the Mandelbrot set with polynomials of degrees of the powers of two.
(May I note that typically on an occasional presentation I just usually give the function series formally and then let the audience discover what is going on—while watching the series of these polynomials develop.)
After all this, let us see a few more interesting functions in the section Favourites.