WOLFRAM|DEMONSTRATIONS PROJECT

3D Plots of Rational Functions of a Complex Variable

​
m
1
2
3
4
5
6
7
m
2
1
2
3
4
m
3
1
2
3
m
4
0
1
n
1
2
3
4
5
6
7
n
2
1
2
3
4
n
3
1
2
3
n
4
0
1
plot points × 20
1
2
3
4
5
plot range
0.5
1
2
3
4
axes
a
2
a
3
a
4
b
2
b
3
b
4
5
z
-0.1
2
z
7
z
-0.1z+(1+)
This Demonstration shows a complex rational function
f(z)
as a 3D plot of
|f(x)|
, in which colors depend on the quadrant in which
f(z)
falls. A rational function is the quotient of two polynomials,
P(z)
and
Q(z)
. This Demonstration uses polynomials of the form
P(z)=
m
1
z
+
a
2
m
2
z
+
a
3
m
3
z
+
a
4
m
4
z
and
Q(z)=
n
1
z
+
b
2
n
2
z
+
b
3
n
3
z
+
b
4
n
4
z
, where the coefficients
a
k
and
b
k
are complex numbers. Suppose that
P(z)
and
Q(z)
have no common roots. Then the zeros of
P(z)
are the zeros of
f(z)
, while the zeros of
Q(z)
are the poles of
f(z)
, which show as tubes rising to infinity.