Zeros, Poles, and Essential Singularities
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 , .[more]
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 (, 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 .[less]
 A. Sveshnikov and A. Tikhonov, The Theory of Functions of a Complex Variable (G. Yankovsky, trans.), Moscow: Mir Publishers, 1971.