# Zeros, Poles, and Essential Singularities

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 , .

f:

z

w=f(z)

r

g

b

o

|w|

|w|>a

a

Re(w)>0

r=1

Im(w)Re(w)<0

g=1

Re(w)<0

b=1

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

w=z

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 .

z=0

n

z

-n

z

-n

z

e

n

n

In the case of a pole, , as .

|w|∞

z0

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 .

ϵ>0

z

0

f(z)

z

1

Z

ϵ