WOLFRAM|DEMONSTRATIONS PROJECT

Continuity of Polynomials in the Complex Plane

​
multiplicity of zero
n
1
1
n
2
2
n
3
3
plot range
0.1
0.2
0.5
1
2
4
resolution
1
2
3
4
axes
continuity
ϵ
0
δ
0.3
×
This Demonstration shows polynomials with zeros (or roots) of varying multiplicity in the complex plane. The object is to show that such a polynomial is a continuous function at a selected point
z
0
.
Let
f:EC
be a complex-valued function, where
E⊂C
. The function
f(z)
is continuous at a point
z
0
if for every
ϵ>0
there is a
δ>0
such that for all points
z∈E
that satisfy the inequality
|z-
z
0
|<δ
, the inequality
f(z)-f(
z
0
)<ϵ
holds.
Assign a color to each point
z
of the complex plane as a function of
w=f(z)-f(
z
0
)
, namely the RGB color of three arguments
r
,
g
,
b
(red, green, blue). If
|w|<ϵ
(with
ϵ
chosen by slider), use black. Otherwise, if
Re(w)>0
, let
r=1
; if
Im(w)Re(w)<0
, let
g=1
; if
Re(w)<0
, let
b=1
.
The zeros of the polynomial can be set using the three locators, and their respective multiplicities can be selected. The movable white
×
marks the point
z
0
.
A black patch around
z
0
means that
w<ϵ
. Subsequently, finding a
δ
such that the circle
z-
z
0
<δ
is inside the patch verifies continuity.
Use high resolution after setting all the arguments and parameters.