Order of an entire function

Summary

The order of an entire function measures how fast its maximum modulus grows, on the scale of functions $\exp(r^\lambda)$. It is the least exponent $\rho$ such that $\abs{f(z)}$ is at most about $\exp(\abs{z}^{\rho})$ for large $\abs{z}$. Polynomials have order $0$, $e^z$ has order $1$, and $e^{z^2}$ has order $2$. The order controls the number of zeros of the function and the shape of its factorization, as in the Hadamard factorization theorem.