Entire Functions

Entire functions are holomorphic on all of $\mathbb{C}$ . Their global theory, including growth estimates, factorization, and value distribution, reveals deep structural properties.

Growth and Order

Definition9.1Order of an entire function

The order of an entire function $f$ is

$\rho = \limsup_{r\to\infty}\frac{\log\log M(r)}{\log r}$

where $M(r) = \max_{|z|=r}|f(z)|$ . The type (for finite order $\rho$ ) is $\sigma = \limsup_{r\to\infty} \frac{\log M(r)}{r^\rho}$ .

ExampleOrders of common entire functions

Polynomials: $\rho = 0$ .
$e^z$ : $M(r) = e^r$ , so $\rho = 1$ and $\sigma = 1$ (normal type).
$e^{z^2}$ : $M(r) = e^{r^2}$ , so $\rho = 2$ and $\sigma = 1$ .
$\cos z$ : $M(r) \sim e^r/2$ , so $\rho = 1$ and $\sigma = 1$ .
$\sin(\sqrt{z})/\sqrt{z}$ : $\rho = 1/2$ .

Weierstrass Factorization

Definition9.2Elementary factors

The elementary factor of order $p$ is

$E_p(z) = (1-z)\exp\left(z + \frac{z^2}{2} + \cdots + \frac{z^p}{p}\right).$

For $p = 0$ : $E_0(z) = 1 - z$ . The exponential factor ensures convergence of infinite products.

Theorem9.1Weierstrass factorization theorem

Let $f$ be entire with zeros $\{a_n\}$ (repeated by multiplicity), $f(0) \neq 0$ . There exist integers $\{p_n\}$ and an entire function $g$ such that

$f(z) = e^{g(z)}\prod_{n=1}^{\infty}E_{p_n}(z/a_n).$

If the zeros satisfy $\sum |a_n|^{-(p+1)} < \infty$ for some $p$ , we can take all $p_n = p$ , and the product converges. The smallest such $p$ is the genus of the product.

ExampleFactorization of $\sin(\pi z)$

The zeros of $\sin(\pi z)$ are $\{n : n \in \mathbb{Z}\}$ . Since $\sum 1/|n|^2 < \infty$ , we can use $p = 1$ :

$\sin(\pi z) = \pi z \prod_{n=1}^{\infty}\left(1 - \frac{z^2}{n^2}\right) = \pi z \prod_{n=1}^{\infty}E_1(z/n)E_1(-z/n).$

Setting $z = 1/2$ : $1 = \frac{\pi}{2}\prod_{n=1}^\infty(1-1/(4n^2))$ , yielding Wallis's product $\frac{\pi}{2} = \prod_{n=1}^\infty\frac{4n^2}{4n^2-1}$ .

The Hadamard Factorization Theorem

Theorem9.2Hadamard factorization theorem

If $f$ is entire of finite order $\rho$ with zeros $\{a_n\}$ and $f$ vanishes to order $m$ at the origin, then

$f(z) = z^m e^{Q(z)}\prod_{n} E_p(z/a_n)$

where $Q$ is a polynomial of degree at most $\rho$ and $p = \lfloor\rho\rfloor$ .

ExampleHadamard factorization of $\cos(\pi z)$

$\cos(\pi z)$ has order $1$ , zeros at $z = n + 1/2$ for $n \in \mathbb{Z}$ . By Hadamard:

$\cos(\pi z) = e^{az+b}\prod_{n=-\infty}^{\infty}\left(1 - \frac{z}{n+1/2}\right)e^{z/(n+1/2)} = \prod_{n=0}^{\infty}\left(1 - \frac{4z^2}{(2n+1)^2}\right).$

Summary

RemarkStructure of entire functions

The theory of entire functions reveals that:

Every entire function is determined (up to an exponential factor) by its zeros.
The growth rate constrains how many zeros the function can have (genus $\leq$ order).
The Hadamard factorization provides a canonical representation for functions of finite order.