when all singularities are poles

when all singularities are poles

In the parent entry (

http://planetmath.org/ZerosAndPolesOfRationalFunction

) we see that a

rational function

has as its only singularities a

finite set

of poles. It is also valid the

converse

Theorem

.

Any

single-valued

analytic function

, which has in the whole

closed complex plane

no other singularities than poles, is a rational function.

Proof.

Suppose that

z

↦

w

⁢

(

z

)

is such an analytic function. The number of the poles of

w

must be finite, since otherwise the set of the poles would have in the closed complex plane an

accumulation point

which is neither a point of

regularity

(

http://planetmath.org/

Holomorphic

) nor a pole. Let

b

1

,

b

2

,

…

,

b

k

and possibly

∞

be the poles of the

function

w

.

For every

i

=

1

,

2

,

…

,

k

,  the function has at the pole

b

i

with the order

n

i

, the

Laurent expansion

of the form

w

⁢

(

z

)

=

c

-

n

i

(

z

-

b

i

)

n

i

+

c

-

n

i

+

1

(

z

-

b

i

)

n

i

-

1

+

…

+

c

0

+

c

1

⁢

(

z

-

b

i

)

+

…

(1)

This is in in the greatest open disc containing no other poles. We write (1) as

w

⁢

(

z

)

=

F

n

i

⁢

(

1

z

-

b

i

)

+

P

⁢

(

z

-

b

i

)

,

(2)

where the first addend is the

principal part

of (1), i.e. consists of the terms of (1) which become

infinite

in

z

=

b

i

.

If we think a circle having center in the origin and containing all the finite poles

b

i

(an

annulus

ϱ

<

|

z

|

<

∞

), then

w

⁢

(

z

)

has outside it the

Laurent series

expansion

w

⁢

(

z

)

=

d

m

⁢

z

m

+

d

m

-

1

⁢

z

m

-

1

+

…

+

d

0

+

d

-

1

z

+

…

,

which we write, corresponding to (2), as

w

⁢

(

z

)

=

G

m

⁢

(

z

)

+

Q

⁢

(

1

z

)

,

(3)

where

G

m

⁢

(

z

)

is a

polynomial

of

z

and

Q

⁢

(

1

z

)

a

power series

in

1

z

. Then the equation

R

⁢

(

z

)

:=

∑

i

=

1

k

F

n

i

⁢

(

1

z

-

b

i

)

+

G

m

⁢

(

z

)

defines a rational function having the same poles as

w

. Therefore the function defined by

f

⁢

(

z

)

:=

w

⁢

(

z

)

-

R

⁢

(

z

)

is

analytic

(

http://planetmath.org/Analytic

) everywhere except possibly at the points

z

=

b

i

and

z

=

∞

.  If we write

f

⁢

(

z

)

=

[

w

⁢

(

z

)

-

F

n

i

⁢

(

1

z

-

b

i

)

]

-

∑

j

≠

i

F

n

j

⁢

(

1

z

-

b

j

)

-

G

m

⁢

(

z

)

,

we see that

f

⁢

(

z

)

is

bounded

in a

neighbourhood

of the point

b

i

and is analytic also in this point (

i

=

1

,

2

,

…

,

k

). But then again, the

f

⁢

(

z

)

=

[

w

⁢

(

z

)

-

G

m

⁢

(

z

)

]

-

∑

j

=

1

k

F

n

j

⁢

(

1

z

-

b

j

)

shows that

f

is analytic in the

infinity

(

http://planetmath.org/RiemannSphere

), too. Thus

f

is analytic in the whole closed complex plane. By Liouville’s theorem,

f

is a

constant function

.  We conclude that

R

⁢

(

z

)

+

f

⁢

(

z

)

=

w

⁢

(

z

)

is a rational function. Q.E.D.

The theorem implies, that if a

meromorphic function

is

regular at infinity

or has there a pole, then it is a rational function.

References

1

R. Nevanlinna & V. Paatero

:

Funktioteoria

.  Kustannusosakeyhtiö Otava, Helsinki (1963).

Title

when all singularities are poles

Canonical name

WhenAllSingularitiesArePoles

Date of creation

2014-11-21 21:30:22

Last modified on

2014-11-21 21:30:22

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Theorem

Classification

msc 30D10

Classification

msc 30C15

Classification

msc 30A99

Related topic

RiemannSphere

Related topic

ZeroesOfAnalyticFunctionsAreIsolated

Related topic

Meromorphic

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