Let

be a complex-valued function. Assign a color to each point

of the complex plane as a function of

, namely the RGB color with four arguments

,

,

, and

(red, green, blue, and opacity, all depending on

). If

(with

chosen by its slider), use black. Otherwise: if

,

; if

,

; if

,

.

For

, this colors the four quadrants red, cyan, blue, and yellow.

To illustrate zeros, poles, and essential singularities, choose

and three kinds of functions

,

, and

. Note the characteristic

-fold symmetry in case of a zero or pole of order

.

In the case of a pole,

, as

.

The following theorem is attributed to Sokhotsky and Weierstrass ([1], p. 116). For any

, in any neighborhood of an essential singularity

of the function

, there will be at least one point

at which the value of the function differs from an arbitrary complex number

by less than

.