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 functionMathworldPlanetmath. 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=1fk(z) converges uniformly and absolutely (http://planetmath.org/AbsoluteConvergenceOfInfiniteProduct) on compact subsets, then it converges to a holomorphic function given that all the fk(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 GC be a domain, let {ak} be a sequence of points in G with no accumulation pointsPlanetmathPlanetmath in G, and let {nk} 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 ak and the order of the pole or zero at ak is nk (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.


We call

E0(z) :=1-z,
Ep(z) :=(1-z)ez+12z2++1pzp  for p1,

an elementary factor.

Now note that for some a\{0}, Ep(z/a) has a zero (zero of order 1) at a.

Theorem (Weierstrass Factorization).

Suppose f be an entire functionMathworldPlanetmath and let {ak} be the zeros of f such that ak0 (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 {pk} such that


Note that we can always choose pk=k-1 and the product above will converge as needed, but we may be able to choose better pk for specific functions.


As an example we can try to factorize the function sin(πz), which has zeros at all the integers. Applying the Weierstrass factorization theorem directly we get that


where g(z) is some holomorphic function. It turns out that eg(z)=π, and rearranging the product we get


This is an example where we could choose the pk=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].


  • 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