Weierstrass Factorization Theorem

Factorization of Entire Functions

Motivation: From Polynomials to Entire Functions

A polynomial $P(z)$ of degree $n$ has exactly $n$ zeros (with multiplicities) and can be decomposed as $P(z) = c(z-z_1)\cdots(z-z_n)$. Can we similarly "decompose" an entire function with infinitely many zeros? The Weierstrass theorem gives an affirmative answer, but with a caveat: to ensure the convergence of the infinite product, "canonical factors" are needed.

Entire Functions

An entire function is one that is holomorphic on all of $\mathbb{C}$. Examples: polynomials, $e^z$, $\sin z$, $\cos z$.

Order of growth: $\rho = \limsup_{r \to \infty} \frac{\ln \ln M(r)}{\ln r}$, where $M(r) = \max_{|z|=r} |f(z)|$.

• Polynomials: $\rho = 0$.
• $e^z$: $M(r) = e^{r}$, $\ln \ln e^r = \ln r$, $\rho = 1$.
• $\sin z$: $\rho = 1$ ($|\sin z| \leq e^{|\operatorname{Im} z|}$).

By Liouville's theorem, a bounded entire function is constant.

Canonical Factor

To ensure the convergence of the infinite product, the canonical factor is introduced:

$ E(u, p) = (1-u) \cdot \exp\left(u + \frac{u^2}{2} + \ldots + \frac{u^p}{p}\right). $

As $u \to 0$: $E(u, p) \to 1$ (no oscillations). At $u=1$: $E(1, p) = 0$ (zero).

Weierstrass Factorization Theorem

Theorem: Any entire function $f$ with zeros ${a_n}$ ($a_1, a_2, \ldots$, multiplicities counted, $0 < |a_1| \leq |a_2| \leq \ldots$) and a zero of order $m$ at $0$ can be represented as:

$ f(z) = z^{m} \cdot e^{g(z)} \cdot \prod_{n=1}^\infty E\left(\frac{z}{a_n}, p_n\right), $

where $g(z)$ is an entire function, and $p_n$ are chosen to ensure convergence.

Euler's Product: $\sin (\pi z) = \pi z \cdot \prod_{n=1}^\infty \left(1 - \frac{z^2}{n^2}\right)$.

Zeros of $\sin(\pi z)$: $z = 0, \pm1, \pm2, \ldots$ Order $m = 1$ (zero at $0$), $E(z^2/n^2, 1) = (1 - z^2/n^2)\cdot e^{z^2/n^2}$? No — for $\sin$ a symmetric form with pairs $\pm n$ is used.

Numerical Example

Problem: Obtain Wallis' formula from Euler's product.

Step 1. Euler's product: $\sin(\pi z) = \pi z \cdot \prod_{n=1}^\infty \left(1 - \frac{z^2}{n^2}\right)$.

Step 2. Substitute $z = 1/2$: $\sin(\pi/2) = 1 = \pi \cdot (1/2) \cdot \prod_{n=1}^\infty \left(1 - \frac{1}{4 n^2}\right)$.

Step 3. $1 = (\pi/2) \cdot \prod_{n=1}^\infty \frac{4 n^2 - 1}{4 n^2} \rightarrow \pi/2 = \prod_{n=1}^\infty \frac{4 n^2}{4 n^2 - 1}$.

Step 4. $\frac{4 n^2}{4 n^2 - 1} = \frac{(2n)^2}{(2n-1)(2n+1)} \rightarrow \frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdots$

Wallis’ formula: $ \frac{\pi}{2} = \prod_{n=1}^\infty \frac{(2n)^2}{(2n-1)(2n+1)} = \frac{2 \cdot 2}{1 \cdot 3} \cdot \frac{4 \cdot 4}{3 \cdot 5} \cdot \frac{6 \cdot 6}{5 \cdot 7} \cdots $

This is the classical infinite formula for $\pi$ (Wallis, 1656), which follows from the factorization of the sine.

Check: Partial product up to $n=3$: $(4/3)\cdot(16/15)\cdot(36/35) = 4\cdot16\cdot36/(3\cdot15\cdot35) = 2304/1575 \approx 1.463$. $\pi/2 \approx 1.5708$. The convergence is slow—many factors are needed.

Real Application

Signal theory: filters with zeros at given frequencies are built through the product of factors $(z - z_n)$ — a direct analogue of the Weierstrass factorization. Infinite products arise in the theory of modular forms and algebraic geometry (Dedekind’s $\eta$-function).

Additional Aspects

Entire functions—holomorphic on the whole plane—are classified by their growth. The order $\rho = \limsup \ln \ln M(r)/\ln r$ and type $\sigma$ describe how fast $|f(z)|$ grows as $|z| \to \infty$. Hadamard’s theorem represents an entire function of finite order as a product over its zeros with an exponential factor, which gives a powerful tool in number theory (the Hadamard–Weierstrass product for the $\zeta$-function). The Laplace transform $Lf = \int_0^\infty f(t)e^{-st}dt$ connects functions of a real argument to holomorphic functions in the half-plane $\mathrm{Re}, s > \sigma$; the inverse transform is expressed as a contour integral along a vertical line and is solved using the residue theorem — this is a basic tool in control theory and the analysis of linear dynamical systems.

The combination of entire functions of finite order and the Laplace transform covers most applied problems of linear dynamics: stability, long-term asymptotics, and transients are expressed through the location of the poles and the growth of the entire part of the transform. This is the basic language for automatic control theory, time series analysis, and numerical methods for solving linear differential equations.

Connection with Other Branches of Mathematics

Factorization of entire functions naturally appears in the spectral theory of linear operators. For differential operators with constant coefficients, the characteristic polynomial leads to exponential solutions, while for operators with analytic potential the analogue is the entire characteristic function (for example, the Fredholm determinant), whose zeros describe the spectrum. In the works of Carleman and de Bruijn, factorization is used in the study of entire functions of exponential type and related Paley–Wiener type inequalities.

In the theory of differential equations, direct applications are found in the factorization of solutions to special equations such as the Bessel equation and the Gaussian equation. Classical products for Bessel functions (Hankel, Neumann) and the gamma function (Euler, Gauss) are regarded as particular cases of Weierstrass products. Through these, the locations of eigenfrequencies in Sturm–Liouville problems are described, and asymptotics are constructed via the zeros.

In algebraic geometry, the analogue is the decomposition of holomorphic sections of line bundles by their divisors of zeros. The Riemann–Roch theorem and the Abel–Jacobi construction essentially generalize factorization: the linear space of meromorphic functions with a given divisor is described in terms of linear constraints on zeros and poles.

In analytic number theory, the factorization of entire functions is at the foundation of Dirichlet products. The Hadamard–Weierstrass product for the Riemann zeta function (Hadamard, de la Vallée Poussin) relates the distribution of zeros to the asymptotics of $\pi(x)$. Later, similar constructions were used by Selberg and Gelfand–Shilov in the study of $L$-functions and automorphic forms.

Numerical methods rely on factorization in the construction of stable algorithms for computing special functions: products over zeros allow control of errors for large arguments and avoid catastrophic cancellation, forming the basis of libraries like Abramowitz–Stegun and modern packages such as mpmath.

Historical Background and Development of the Idea

The first prototypes of factorization appeared in the 18th century in the works of Euler: his product for the sine (1738) and the gamma function (the 1730s, published in Commentarii Academiae Scientiarum Petropolitanae) demonstrated that the placement of zeros and poles almost uniquely determines a function. However, the general theory for arbitrary entire functions was formulated only by K. Weierstrass in 1876 in an article in the Journal für die reine und angewandte Mathematik. The motivation was the problem of constructing analytic functions with prescribed zeros and representing periodic functions as products.