the order read from the Taylor coefficients

An entire function carries the same information in two forms: the values it takes across the plane, and the list of Taylor coefficients a_0, a_1, a_2, ... in its power series sum a_n z^n. We measured growth using the values, through M(r). It is a small marvel that the very same growth — the order and the type — can be read directly off the coefficients, without ever looking at the function out at large radius. How fast the coefficients shrink to zero secretly encodes how fast the function grows to infinity.

The rule: an entire function f(z) = sum a_n z^n has order rho given by rho = limsup as n tends to infinity of (n log n) / (log(1 / |a_n|)). The faster the coefficients decay, the larger the denominator log(1/|a_n|), and the smaller the order — which fits intuition: tiny coefficients build a tamer function. There is a matching formula for the type once rho is known. A worked check on e^z = sum z^n / n!: here a_n = 1/n!, so 1/|a_n| = n!, and log(n!) is about n log n by Stirling's approximation. Then (n log n) / (n log n) tends to 1, giving order rho = 1 — exactly right.

This coefficient route is the practical way to determine the order when you only have the series, which is common: you might define a function by a power series and need its order to apply Hadamard or to know how many zeros it can have. It connects the analytic notion of growth to the algebraic data of the coefficients. The honest caution: the formula is a limsup over n, so a few unusually large or small coefficients can mislead a quick glance — the order is governed by the long-run trend of how |a_n| behaves, not by any single term.