Square
We demonstrate the complex square function using various colorings.
\(f(z) = z^2,\quad z \in \mathbb{C}(0,3)\)
Recall that squaring a complex number results in multiplying its phase by two and squaring its magnitude.
This function is not injective: each color can be found twice on the plot. In fact the whole complex plane is produced as the image of the right halfplane (with real parts positive) by the square function: we can found every color of the coloring on the right hand side of the plot. (Similar can be stated—among others—concerning the left halfplane too.) We could also observe that because the square of a number and its additive inverse is the same: the figure is point symmetric.
\(f(z) = z^2,\quad z \in \mathbb{C}(0,3)\)
The previous sentences can be verified on this picture too.
The square function has a zero of multiplicity two at zero. Note that when going round the the origin (the root in general) the colors repeat twice. (Linear functions, the identity function included, have one zero of multiplicity one. See e.g. the images of the colorings.) This will be a nice method to obtain the multiplicity of a zero.
It is interesting to observe the plot of the square function using a coloring with 'stairs':
\(f(z) = z^2,\quad z \in \mathbb{C}(0,3)\)
The resulting lines articulate locations with function values of constant real (or imaginary) parts. It can be shown that these are hyperbolic curves. Not to be confused with the images of lines parallel to the real (or imaginary) axis which are parabolae.
These can be confirmed looking at the calculation \[ z^2 = (x+yi)^2 = x^2 - y^2 + 2xy \cdot i \] in different ways…