Complex quadratic polynomial

A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.

Forms

When the quadratic polynomial has only one variable (univariate), one can distinguish its 4 main forms:

The general form:

f(x) = a_2 x^2 + a_1 x + a_0 \qquad \,

where

\qquad a_2 \ne 0

The factored form used for logistic map

f_r(x) = r x ( 1-x ) \,

f_{\theta}(x) = x^2 + e^{2 \pi \theta i} x \,

which has an indifferent fixed point with multiplier

\lambda = e^{2 \pi \theta i} \,

at the origin

The monic and centered form,

f_c(x) = x^2 +c\,

The monic and centered form has the following properties:

It is the simplest form of a nonlinear function with one coefficient (parameter),

It is a centered polynomial (the sum of its critical points is zero),

It can be postcritically finite, i.e. If the orbit of the critical point is finite. It is when critical point is periodic or preperiodic.

Since $f_c(x) \,$ is affine conjugate to the general form of the quadratic polynomial it is often used to study complex dynamics and to create images of Mandelbrot, Julia and Fatou sets.

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