Order Of An Entire FunctionEdit
The order of an entire function is a key concept in complex analysis that measures how rapidly a function grows as its argument moves far away from the origin in the complex plane. If f is an entire function, define M(r) to be the maximum of |f(z)| on the circle of radius r, that is M(r) = max_{|z| = r} |f(z)|. The order ρ is then defined by ρ = limsup_{r → ∞} log log M(r) / log r. When this limit is finite, f is said to be of finite order; if the limit is infinite, f has infinite order. In practice, many classical entire functions have finite order, and their growth can be described in precise terms.
The notion of order sits inside a broader framework that connects growth, zeros, and representation. It provides a coarse but powerful classification: functions of finite order grow at a rate that, while unbounded, is constrained in a way that can be quantified. The order also interacts with the distribution of zeros through factorization theorems, tying growth to the way zeros occur in the complex plane. These ideas have a long history in the development of entire function theory and are foundational in both pure and applied contexts.
Definitions and basic notions
Entire function: A complex function that is analytic on the entire complex plane. See entire function for a general overview and context.
Maximum modulus M(r): The quantity M(r) = max_{|z| = r} |f(z)|, which tracks the largest value of |f| on circles centered at the origin.
Order ρ: The finite-integer or real number ρ = limsup_{r → ∞} log log M(r) / log r. If this limit is finite, f is of finite order; otherwise f has infinite order.
Lower order: The corresponding quantity defined with liminf instead of limsup, giving a slightly weaker notion of growth rate.
Type (within a given order): If f has finite order ρ, one can examine the type by δ = limsup_{r → ∞} log M(r) / r^ρ. The type gives a finer measure of growth inside the same order class.
Zeros and growth: The zeros of an entire function play a central role in its growth behavior. The way zeros are distributed influences the possible order and genus of the function, as reflected in factorization formulas discussed below.
Hadamard factorization context: Functions with specified growth and zero patterns admit products that encode their zeros. See Hadamard factorization theorem for a canonical way to write an entire function in terms of its zeros and an exponential factor.
Examples and intuition
f(z) = e^{a z} with a ≠ 0: This function is entire and has order 1. Its growth is essentially exponential in the horizontal direction, and the type corresponds to |a|.
f(z) = sin z: As a sine, this function is entire and also of order 1, with growth controlled by the exponential components that appear in its complex representation.
f(z) = e^{z^2}: This function has order 2, reflecting its more rapid growth along certain directions in the complex plane.
General polynomials in z and e^{P(z)}: If P is a polynomial of degree d, then e^{P(z)} has order d; multiplying by polynomials does not change the order, though it can affect the type.
These examples illustrate how the order captures a broad spectrum of growth behaviors, from linear to quadratic and beyond, across entire functions built from exponentials and polynomials.
Hadamard factorization and the zeros
One of the central links between growth and zeros is provided by Hadamard’s factorization theorem. For a nonzero entire function f with zeros {a_n} (taken with multiplicities) and, if appropriate, a zero of order m at z = 0, one can often represent f in a form that makes its growth and zero structure explicit:
f(z) = z^m e^{g(z)} ∏_{n} E_p(z / a_n)
Here: - g is a polynomial (or more generally, an entire function of controlled growth), - E_p is a canonical product factor associated with the zero at a_n, defined by E_p(w) = (1 - w) exp(w + w^2/2 + ... + w^p/p), - p is a nonnegative integer chosen so that the product converges (this p is called the genus of the function).
A key quantity is the exponent of convergence of the zeros, defined as the infimum of p such that ∑ |a_n|^{−p} converges. There is a deep connection between this exponent and the order ρ: roughly speaking, the zeros cannot be too dense if the function is of finite order, and the exponent of convergence of the zeros is bounded above by the order. This interplay is a central theme in the study of entire functions and their growth.
- The Hadamard factorization theorem itself provides a constructive way to rebuild an entire function from its zeros and a controlled exponential factor, tying together the growth rate (order and type) with the zero distribution. See Hadamard factorization theorem for a detailed statement and proofs.
Growth, order, and zero distribution
Finite order and zero distribution: If f has finite order ρ, then the zeros of f are constrained in a way that prevents excessive clustering in too small regions of the plane. Conversely, the way zeros accumulate can force a given growth rate, linking order to the exponent of convergence of zeros.
Connections to broader theory: The study of order is part of a larger framework that includes growth estimates, canonical products, and factorization techniques. These ideas also connect to topics like the distribution of values of entire functions and, in more advanced settings, Nevanlinna theory, which analyzes how often an entire function can attain given values.
Practical uses: Understanding the order of an entire function helps in approximation problems, in the design of functions with prescribed growth, and in the qualitative study of differential equations in which entire functions appear as solutions or generating functions.