Functions…
On the following pages we shall found the plot of a complex function in the layout below. (Here only the placeholders are shown.)
\(f(z) = \dots,\quad z \in \mathbb{C}(z_0,r)\)
In the place of the big white square we shall find the plot itself, and the little white square in the upper right corner shall contain the coloring applied—as introduced on the previous page.
Under the picture we will give the formula of the function and also the domain. To enable us to speak about square domains like this in a simple manner, let us introduce the following notation: \[\mathbb{C}(z_0,r) := \left\{\, z \in \mathbb{C} : \left| \Re (z - z_0) \right| < r, \left| \Im (z - z_0) \right| < r \,\right\},\] where we give the center and the 'radius' of the domain. So the length of every edge is 2r units.
Tipically we will choose a neighbourhood of zero with a small integer number as radius.