# Plotting Rational Functions of a Complex Variable

Plotting Rational Functions of a Complex Variable

This Demonstration shows a complex rational function as a contour plot superposed on a parametric plot, in which colors depend on the quadrant in which falls. A rational function is the quotient of two polynomials, and . This Demonstration uses 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 the centers of the black patches, and poles are shown as black points.

f(z)

f(z)

P(z)

Q(z)

P(z)=+++

m

1

z

a

2

m

2

z

a

3

m

3

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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

k

b

k

P(z)

Q(z)

P(z)

f(z)

Q(z)

f(z)