the order of an entire function
/ order, rho is spelled 'roh' /
Entire functions come in very different sizes. A polynomial barely grows; e^z grows fast; e^(z^2) grows ferociously fast; e^(e^z) is beyond reckoning. We want a single number that places each one on this scale of explosiveness. That number is the order. It answers the question 'when I compare f's growth to a clean benchmark e^(r^rho), which exponent rho is the dividing line between functions that grow slower and functions that grow faster?'
The benchmark is the double-exponential expression e^(r^rho). The order rho of an entire function f is the smallest number such that M(r) is eventually bounded above by e^(r^(rho + epsilon)) for every epsilon > 0. As a formula: rho = limsup as r tends to infinity of log log M(r) / log r. The double logarithm is the trick — taking log once turns M(r) of size e^(r^rho) into r^rho, and the second log turns that into rho log r, so dividing by log r isolates rho. Worked values: every polynomial has order 0; e^z has order 1 (since M(r) = e^r, log log M(r) = log r, ratio 1); e^(z^2) has order 2; cos(sqrt z), which is entire, has order 1/2. The order need not be an integer.
Order is the master invariant of an entire function's growth, and almost every deep theorem in this field is stated in terms of it. Finite order is the hypothesis under which Hadamard's factorization works and under which the zeros cannot be too dense; the order ties the growth of f directly to the density of its zeros. One honest subtlety: the order is defined by a limsup, so it captures the worst, fastest stretch of growth, not a steady rate; a function can have order 1 while growing like e^r along some sequence of radii and far slower along others.
Compute the order of f(z) = e^(z^3). Here M(r) = e^(r^3) along the rays where Re(z^3) = r^3, so log M(r) = r^3, log log M(r) = 3 log r, and the ratio with log r is 3. The order is 3 — exactly the degree of the polynomial in the exponent, which is the general rule for f = e^P(z).
For f = e^P(z) with P a polynomial, the order equals the degree of P. The double log peels back exactly to that exponent.
The order is a limsup, so it records the fastest growth that ever occurs, not a uniform rate; two functions of the same order can behave very differently between radii. And finite order is a real restriction — e^(e^z) has infinite order and escapes the whole theory.