Rational Functions with Complex Coefficients
Rational Functions with Complex Coefficients
A quotient of two polynomials and , , is called a rational function. This Demonstration plots rational functions using polynomials of the form and , where the coefficients and are complex numbers. Suppose that and have no common roots. Then the zeros of are the zeros of , and the zeros of are the poles of . Zeros are shown in white in centers of black patches, and poles are shown as red points.
P(z)
Q(z)
R(z)=P(z)/Q(z)
P(z)=+++
m
1
z
a
2
m
2
z
a
3
m
3
z
a
4
m
4
z
Q(z)=+++
n
1
z
b
2
n
2
z
b
3
n
3
z
b
4
n
4
z
a
i
b
i
P(z)
Q(z)
P(z)
R(z)
Q(z)
R(z)