nLab Weierstrass factorization theorem

Idea

The analogue of polynomial factorization of polynomial functions over the complex numbers, but for entire complex analytic functions, which come with a countable set of roots instead of a finite set of roots for polynomials.

Definition

The Weierstrass primary factors are a sequence of functions on the complex functions inductively defined as

$$E_0(z) = (1 - z)$$

$$E_{n + 1}(z) = E_{n}(z) e^{\frac{z^{n + 1}}{n + 1}}$$

Given a complex analytic function $f$ which is entire in the sense that the Taylor series expansion of $f$ converges on all of the complex numbers, and which has a zero of multiplicity $n$ at $z = 0$, the Weierstrass factorization theorem states that there exists a countable set of roots $z_i$ with index set $I$ such that

$$f(z) = z^n e^{g(z)} \prod_{i \in I} E_{p_i}\left(\frac{z}{z_i}\right) \mathrlap{\,.}$$

where $n$ is a natural number, $g(z)$ is an entire function and $E_{p_i}$ are Weierstrass primary factors and $p_i$ are natural numbers chosen to ensure the infinite product converges.

Examples

Polynomial functions

For a polynomial function $f$ of degree $n$ with $m \leq n$ zeros at $z = 0$, which is always an entire function, the countable set $I$ for roots $z_i$ is a finite set of cardinality $n - m$, the entire function $g(z)$ is equal to a constant $c$, and the Weierstrass primary functions used in the product is $E_{0}(z)$. Thus, Weierstrass factorization can be expressed as

$$f(z) = z^m e^c \prod_{i = 1}^{n - m} E_{0}\left(\frac{z}{z_i}\right) = c^\prime z^m \prod_{i = 1}^{n - m} (z - z_i) \quad \mathrm{where} \quad c^\prime = e^c \prod_{i = 1}^{n - m} \frac{(-1)^{i}}{z_i} \mathrlap{\,.}$$

which is precisely the fundamental theorem of algebra for complex polynomial functions.

In constructive mathematics

In constructive mathematics, the usual Weierstrass factorization theorem fails for the same reason that the fundamental theorem of algebra fails: the complex numbers are not a discrete field. As a result, there are a few alternatives for the Weierstrass factorization theorem a constructive mathematics.

Approximate Weierstrass factorization theorem

The approximate version of the Weierstrass factorization theorem states that given an entire complex analytic function $f$ with a zero of multiplicity $n$ at $z = 0$, for all positive rational numbers $\epsilon$ one can construct a countable set $I$ of complex numbers $z_i$ such that

$$\vert f(z) - z^n e^{g(z)} \prod_{i \in I} E_{p_i}\left(\frac{z}{z_i}\right) \vert \lt \epsilon \mathrlap{\,.}$$

where $n$ is a natural number, $g(z)$ is an entire function and $E_{p_i}$ are Weierstrass primary factors and $p_i$ are natural numbers chosen to ensure the infinite product converges.

Using multisets of roots

Similarly to complex polynomials, instead of considering individual complex roots, one can instead consider multisets of complex roots for entire functions.

Related concepts

• algebraic closure

• fundamental theorem of algebra

References

See also:

• Wikipedia: Weierstrass factorization theorem