zeros and poles of rational function

zeros and poles of rational function

A

rational function

of a

complex variable

z

may be presented by the equation

R

⁢

(

z

)

=

a

0

⁢

z

m

+

a

1

⁢

z

m

-

1

+

…

+

a

m

b

0

⁢

z

n

+

b

1

⁢

z

n

-

1

+

…

+

b

n

,

(1)

where the numerator and the denominator are mutually

irreducible polynomials

with

complex

coefficients

a

j

and

b

k

(

a

0

⁢

b

0

≠

0

).  If

z

=

x

+

i

⁢

y

(

x

,

y

∈

ℝ

), then the real and

imaginary parts

of

R

⁢

(

z

)

are rational functions of

x

and

y

.

When we factorize the numerator and the denominator in the ring

ℂ

⁢

[

z

]

, we can write

R

⁢

(

z

)

=

a

0

⁢

(

z

-

α

1

)

μ

1

⁢

(

z

-

α

2

)

μ

2

⁢

…

⁢

(

z

-

α

r

)

μ

r

b

0

⁢

(

z

-

β

1

)

ν

1

⁢

(

z

-

β

2

)

ν

2

⁢

…

⁢

(

z

-

β

s

)

ν

s

,

(2)

where

α

j

≠

β

k

for all

j

,

k

.

The form (2) of the rational function expresses the zeros

α

j

and the

infinity

places

β

k

of the

function

.  One can write (2) as

R

⁢

(

z

)

=

(

z

-

α

j

)

μ

j

⁢

S

j

⁢

(

z

)

where

S

j

⁢

(

z

)

is a rational function which in

z

=

α

j

gets a finite non-zero value.  Accordingly one says that the point

α

j

is a

zero

of

R

⁢

(

z

)

with the order

μ

j

(

j

=

1

,

2

,

…

,

r

).  One can also write (2) as

R

⁢

(

z

)

=

1

(

z

-

β

k

)

ν

k

⁢

T

k

⁢

(

z

)

where

T

k

⁢

(

z

)

is a rational function getting in the point

β

k

a finite non-zero value.
As

z

→

β

k

,  the

modulus

|

R

⁢

(

z

)

|

increases unboundedly in such a manner that

|

z

-

β

k

|

ν

k

⁢

|

R

⁢

(

z

)

|

tends to a finite non-zero limit.  So one says that

R

⁢

(

z

)

has in the point

β

k

a

pole

with the order

ν

k

(

k

=

1

,

2

,

…

,

s

).

Behaviour at infinity

Now let

|

z

|

increase unboundedly.  When we write

R

⁢

(

z

)

=

z

m

-

n

⋅

a

0

+

a

1

z

+

…

+

a

m

z

m

b

0

+

b

1

z

+

…

+

b

n

z

n

,

we get three cases:

•

If

m

>

n

,  then

lim

z

→

∞

⁡

R

⁢

(

z

)

=

∞

.  Since

lim

z

→

∞

⁡

R

⁢

(

z

)

z

m

-

n

=

a

0

b

0

is finite and non-zero, the point

z

=

∞

is the pole of

R

⁢

(

z

)

with the order

m

-

n

.

•

If

m

=

n

,  we have

lim

z

→

∞

⁡

R

⁢

(

z

)

=

a

0

b

0

and thus

R

⁢

(

z

)

has in the infinity a finite non-zero value.

•

If

m

<

n

,  we have

lim

z

→

∞

⁡

R

⁢

(

z

)

=

0

in such a manner that

lim

z

→

∞

⁡

z

n

-

m

⁢

R

⁢

(

z

)

=

a

0

b

0

.  This means that

R

⁢

(

z

)

has in infinity a zero with the order

n

-

m

.

In any case,

R

⁢

(

z

)

has equally many zeros and poles, provided that each zero and pole is counted so many times as its order says.  The common number of the zeros and poles is called the

order of the rational function

.  It is the greatest of the degrees (

http://planetmath.org/PolynomialRing

)

m

and

n

of the numerator and denominator.

c

-places

Denote by

c

any non-zero

complex number

.  The

c

-

place

of

R

⁢

(

z

)

means such a point

z

for which

R

⁢

(

z

)

=

c

.  If

z

0

is a

c

-place of

R

⁢

(

z

)

=

P

⁢

(

z

)

Q

⁢

(

z

)

where the

polynomials

P

⁢

(

z

)

and

Q

⁢

(

z

)

have no common factor (

http://planetmath.org/DivisibilityInRings

), then

z

0

is a zero of

R

⁢

(

z

)

-

c

=

P

⁢

(

z

)

-

c

⁢

Q

⁢

(

z

)

Q

⁢

(

z

)

.

(3)

If this zero is of order

μ

, then one says that

z

0

is of order

μ

as the

c

-place of

R

⁢

(

z

)

.  The numerator and denominator of (3) cannot have common factor (otherwise any common factor would be also a factor of

P

⁢

(

z

)

).  This implies that the order of the rational function defined by (3) is the same as the order

k

of

R

⁢

(

z

)

.  Because (3) gets

k

times the value

0

, also

R

⁢

(

z

)

gets

k

times the value

c

.  Thus we have derived the

Theorem.

A rational function attains any complex value so many times as its order is.

References

1

R. Nevanlinna & V. Paatero

:

Funktioteoria

.  Kustannusosakeyhtiö Otava. Helsinki (1963).

Title

zeros and poles of rational function

Canonical name

ZerosAndPolesOfRationalFunction

Date of creation

2014-02-23 18:12:33

Last modified on

2014-02-23 18:12:33

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Topic

Classification

msc 30D10

Classification

msc 30C15

Classification

msc 30A99

Classification

msc 26C15

Related topic

MinimalAndMaximalNumber

Related topic

OrderValuation

Related topic

RolfNevanlinna

Related topic

PlacesOfHolomorphicFunction

Related topic

ZeroOfPolynomial

Defines

order of rational function

Defines

order

Defines

c-place

Defines

place

Generated on Fri Feb 9 20:17:52 2018 by

LaTeXML