Weierstrass factorization theorem

Weierstrass factorization theorem

There are several different statements of this theorem, but in essence this theorem will allow us to prescribe zeros and their orders of a

holomorphic function

. It also allows us to factor any holomorphic function into
a

product

of zeros and a non-zero holomorphic function. We will need to know
here how an infinite product

converges

. It can then be shown
that if

∏

k

=

1

∞

f

k

⁢

(

z

)

converges uniformly

and absolutely (

http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct

)
on

compact subsets

, then it converges to a holomorphic function given that all the

f

k

⁢

(

z

)

are holomorphic. This is what we will by the infinite product in what follows.

Note that once we can prescribe zeros of a

function

then we can also prescribe the poles as well and get a

meromorphic function

just by dividing
two holomorphic functions

f

/

h

where

f

will contribute zeros, and

h

will make poles at the points where

h

⁢

(

z

)

=

0

. So let’s start with the existence statement.

Theorem

(Weierstrass Product).

Let

G

⊂

C

be a domain, let

{

a

k

}

be a

sequence

of
points in

G

with no

accumulation points

in

G

, and let

{

n

k

}

be any
sequence of non-zero integers (

positive

or negative).
Then there exists a function

f

meromorphic

in

G

whose poles and zeros are exactly at the points

a

k

and the order of the pole or zero at

a

k

is

n

k

(a positive order stands for zero, negative stands for pole).

Next let’s look at a more specific statement with more . For
one let’s start looking at the whole

complex plane

and further let’s forget about poles for now to make the following formulas simpler.

Definition

.

We call

E

0

⁢

(

z

)

:=

1

-

z

,

E

p

⁢

(

z

)

:=

(

1

-

z

)

⁢

e

z

+

1

2

⁢

z

2

+

⋯

+

1

p

⁢

z

p

for

p

≥

1

,

an

elementary factor

.

Now note that for some

a

∈

ℂ

\

{

0

}

,

E

p

⁢

(

z

/

a

)

has a zero (zero of order 1) at

a

.

Theorem

(Weierstrass Factorization).

Suppose

f

be an

entire function

and let

{

a

k

}

be the zeros of

f

such that

a

k

≠

0

(the non-zero zeros of

f

). Let

m

be the order of
the zero of

f

at

z

=

0

(

m

=

0

if

f

does not have a zero at

z

=

0

). Then
there exists an entire function

g

and a sequence of non-negative
integers

{

p

k

}

such that

f

⁢

(

z

)

=

z

m

⁢

e

g

⁢

(

z

)

⁢

∏

k

=

1

∞

E

p

k

⁢

(

z

a

k

)

.

Note that we can always choose

p

k

=

k

-

1

and the product above will converge as needed, but we may be able to choose better

p

k

for specific functions.

Example

.

As an example we can try to factorize the function

sin

⁡

(

π

⁢

z

)

, which has zeros at all the integers. Applying the Weierstrass factorization theorem directly we get that

sin

⁡

(

π

⁢

z

)

=

z

⁢

e

g

⁢

(

z

)

⁢

∏

k

=

-

∞

,

k

≠

0

∞

(

1

-

z

k

)

⁢

e

z

/

k

,

where

g

⁢

(

z

)

is some holomorphic function. It turns out that

e

g

⁢

(

z

)

=

π

,
and rearranging the product we get

sin

⁡

(

π

⁢

z

)

=

z

⁢

π

⁢

∏

k

=

1

∞

(

1

-

z

2

k

2

)

.

This is an example where we could choose the

p

k

=

1

for all

k

and thus
we could then get rid of the ugly parts of the infinite product. For
calculations in this example see Conway

[

1

]

.

References

1

John B. Conway.
.
Springer-Verlag, New York, New York, 1978.

2

Theodore B. Gamelin.
.
Springer-Verlag, New York, New York, 2001.

Title

Weierstrass factorization theorem

Canonical name

WeierstrassFactorizationTheorem

Date of creation

2013-03-22 14:19:31

Last modified on

2013-03-22 14:19:31

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

8

Author

jirka (4157)

Entry type

Theorem

Classification

msc 30C15

Synonym

Weierstrass product theorem

Related topic

MittagLefflersTheorem

Defines

elementary factor

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