Entire Functions and Growth Order — Mathematics Knowledge System

Overview

Liouville's theorem says a bounded entire function must be constant, so every non-constant entire function is unbounded — but 'unbounded' comes in wildly different flavors, from the mild growth of a polynomial to the explosive growth of e^(e^z). The order of growth ρ of an entire function f is a single real number (or ∞) that measures how fast max|z|=r |f(z)| grows compared to exponentials of powers of r: ρ is the infimum of exponents α such that |f(z)| ≤ exp(|z|^α) eventually. Polynomials have order 0, e^z and sin(z), cos(z) have order 1, and e^(z^n) has order n; functions like e^(e^z) have infinite order. This single number turns out to control deep structural facts, most famously via the Hadamard factorization theorem: an entire function of finite order ρ can be written as a product over its zeros times e^(polynomial of degree ≤ ρ), generalizing how a polynomial factors over its roots — this is the tool that made possible, for instance, Hadamard's product formula for the Riemann zeta function's completed form and proofs related to the prime number theorem.

Intuition

The order ρ answers the question 'how many exponentials tall is this function, as a power of r?' by taking a double logarithm: ln M(r) measures the exponent to which e must be raised to reach M(r), and then ln of THAT measures what power of r that exponent itself grows like. For f(z)=e^z, M(r) = e^r (attained at z=r), so ln M(r) = r and ln ln M(r) = ln r, giving ρ = lim (ln r)/(ln r) = 1. For f(z) = e^(z^n), M(r) = e^(r^n), so ln M(r) = r^n and ln ln M(r) = n ln r, giving ρ = lim (n ln r)/(ln r) = n. Polynomials grow only polynomially (M(r) ~ r^d), so ln M(r) ~ d ln r grows logarithmically, and ln ln M(r) ~ ln ln r grows far slower than ln r, giving ρ=0. The point of Hadamard's factorization theorem is that this crude growth measurement is exactly the right invariant to control how densely a function's zeros can be packed and what 'correction factor' e^g(z) is needed on top of the naive zero-product — finite order caps the polynomial degree of that correction, turning growth control into structural control.

Formal Definition

Definition

For an entire function f, define M(r) = max_{|z|=r} |f(z)|. The order of growth ρ(f) is:

\rho(f) = \limsup_{r\to\infty} \frac{\ln \ln M(r)}{\ln r}

Order of growth (order of an entire function)

\rho(f) = \inf\{\alpha \ge 0 : M(r) \le e^{r^\alpha} \text{ for all sufficiently large } r\}

Equivalent characterization

f(z) = z^m e^{g(z)} \prod_{n} E_{p}\!\left(\frac{z}{a_n}\right), \qquad \deg g \le \rho, \quad p \le \rho

Hadamard factorization (finite order ρ, zeros a_n, elementary factors E_p)

Worked Examples

On |z|=r, |e^z| = e^{Re(z)} is maximized when Re(z)=r, i.e. z=r, giving M(r) = e^r.
$M(r) = \max_{|z|=r} |e^z| = e^r$
Take the double logarithm: ln M(r) = r, and ln ln M(r) = ln r.
$\ln M(r) = r, \qquad \ln \ln M(r) = \ln r$
The order is the limit of ln ln M(r) / ln r as r → ∞.
$\rho = \lim_{r\to\infty} \frac{\ln r}{\ln r} = 1$

Answer: e^z has order of growth ρ = 1.

Practice Problems

Difficulty 6/10

What is the order of growth of sin(z)? (Hint: for z=iy on the imaginary axis, sin(iy) = i·sinh(y), and sinh(y) ~ e^y/2 for large y.)

Difficulty 7/10

What is the order of growth of f(z) = e^(z^5)?

Difficulty 8/10

A physicist claims a certain entire function f satisfies |f(z)| ≤ C·e^{|z|^{1.5}} for all sufficiently large |z|, for some constant C. Based on this, what can you say about the order of growth of f, and what does that bound on order imply via Hadamard factorization about any polynomial 'correction factor' e^{g(z)} in f's factorization (assuming f has finitely or infinitely many zeros)?

Quiz

Summary

Every non-constant entire function is unbounded (Liouville), but the order of growth ρ measures how fast, via ρ = limsup ln ln M(r) / ln r.
Polynomials have order 0; e^z, sin z, and cos z have order 1; e^(z^n) has order n; some functions (like e^(e^z)) have infinite order.
Finite order caps the structure of an entire function via the Hadamard factorization theorem: f(z) = z^m e^{g(z)} times a canonical product over its zeros, with deg g ≤ ρ.
Growth order connects analytic behavior to algebraic/structural facts, underlying results like Hadamard's product formula used in the study of the Riemann zeta function.

References

WebsiteWikipedia — Entire function
WebsiteWikipedia — Hadamard factorization theorem