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