Weierstrass Factorization Theorem — Mathematics Knowledge System

Overview

The Weierstrass Factorization Theorem generalizes the fact that a polynomial is determined (up to a constant) by its zeros: every entire function can be written as a (possibly infinite) product over its zeros, multiplied by a nonvanishing entire factor e^{g(z)}. The key tool is the Weierstrass elementary factor E_p(z) = (1−z) exp(z + z²/2 + … + z^p/p), which is designed to have exactly one zero at z=1 while converging absolutely in infinite products. This theorem is the foundation for representing classical functions like sin(πz) and the gamma function as infinite products.

Intuition

Just as a polynomial P(z) = c(z−r₁)(z−r₂)⋯(z−rₙ) is built from its roots, an entire function is 'built' from its zeros as an infinite product. The complication is convergence: a raw product ∏(1−z/zₙ) may diverge. Weierstrass's insight is to multiply each factor by a convergence-fixing exponential 'correction' that doesn't introduce new zeros. The elementary factor E_p(z/zₙ) is equal to 1 minus a zero at z=zₙ times a carefully chosen Taylor-exponential correction whose terms cancel the bad parts of log(1−z/zₙ).

Formal Definition

Definition

Let f be an entire function with zeros z₁, z₂, … (listed with multiplicity, 0 excluded separately). The Weierstrass elementary factors are:

E_0(z) = 1-z,\quad E_p(z) = (1-z)\exp\!\left(z + \frac{z^2}{2} + \cdots + \frac{z^p}{p}\right)

Elementary factors

f(z) = z^m e^{g(z)} \prod_{n=1}^{\infty} E_{p_n}\!\left(\frac{z}{z_n}\right)

Weierstrass factorization

\sin(\pi z) = \pi z \prod_{n=1}^{\infty}\left(1 - \frac{z^2}{n^2}\right)

Classic example: sine product

Worked Examples

sin(πz) is entire with zeros exactly at all integers: z = 0, ±1, ±2, …
$\sin(\pi z) = 0 \iff z \in \mathbb{Z}$
Pair up the zeros ±n into factors (1 − z/n)(1 + z/n) = 1 − z²/n².
$\left(1 - \frac{z}{n}\right)\left(1 + \frac{z}{n}\right) = 1 - \frac{z^2}{n^2}$
The series ∑ 1/n² converges, so the product ∏(1 − z²/n²) converges absolutely and uniformly on compact sets (p=1 suffices for the paired factors).
$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} < \infty$
The nonvanishing entire factor is e^{g(z)} where g(z) can be determined by comparing the logarithmic derivatives; for sine it works out to g(z) = log π (a constant), giving the factor π.
$\sin(\pi z) = \pi z \prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)$

Answer: The product converges because ∑1/n² < ∞, and the nonvanishing factor is π (a constant), giving the classical sine product formula.

Practice Problems

Difficulty 7/10

Write the Weierstrass product for cos(πz/2) using its zeros at odd integers.

Difficulty 8/10

What is the Weierstrass product for 1/Γ(z) (the reciprocal gamma function)?

Difficulty 9/10

Using the product formula for sin(πz), derive the Basel sum ∑_{n=1}^∞ 1/n² = π²/6.

Common Mistakes

Common Mistake

The nonvanishing factor e^{g(z)} can always be taken as a constant.

For functions of infinite order, g(z) may be a polynomial or even an entire function of higher growth.

Common Mistake

The product ∏(1 − z/zₙ) always converges.

Without the exponential correction factors, the product diverges if ∑1/|zₙ| = ∞.

Quiz

What is the purpose of the exponential factor in E_p(z)?

To introduce extra zerosTo ensure convergence by canceling leading Taylor terms of log(1−z)To make the product real-valuedTo normalize the function at z=1

The Weierstrass theorem applies to:

Meromorphic functions onlyPolynomials onlyAll entire functionsFunctions on a bounded domain

Summary

Every entire function factors as z^m e^{g(z)} ∏ E_{pₙ}(z/zₙ) where the product runs over nonzero zeros.
Elementary factors E_p(z) = (1−z) exp(z + z²/2 + … + z^p/p) ensure convergence.
The sine product sin(πz) = πz ∏(1 − z²/n²) is the canonical example.
Comparing Taylor coefficients in the sine product yields the Basel sum π²/6.
The reciprocal gamma function 1/Γ(z) is also given by a Weierstrass product.

References

WebsiteWikipedia — Weierstrass factorization theorem