Order and Growth
The order (at infinity) of an entire function f(z) is defined using the limit superior as:
where Br is the disk of radius r and denotes the supremum norm of f(z) on Br. If 0<ρ<∞, one can also define the type:
In other words, the order of f(z) is the infimum of all m such that f(z) = O(exp(|z|m)) as z → ∞. The order need not be finite.
Entire functions may grow as fast as any increasing function: for any increasing function g: [0,∞) → R there exists an entire function f(z) such that f(x)>g(|x|) for all real x. Such a function f may be easily found of the form:
- ,
for a conveniently chosen strictly increasing sequence of positive integers nk. Any such sequence defines an entire series f(z); and if it is conveniently chosen, the inequality f(x)>g(|x|) also holds, for all real x.
Read more about this topic: Entire Function
Famous quotes containing the words order and, order and/or growth:
“However fiercely opposed one may be to the present order, an old respect for the idea of order itself often prevents people from distinguishing between order and those who stand for order, and leads them in practise to respect individuals under the pretext of respecting order itself.”
—Antonin Artaud (1896–1948)
“The herd of mankind can hardly be said to think; their notions are almost all adoptive; and, in general, I believe it is better that it should be so; as such common prejudices contribute more to order and quiet, than their own separate reasonings would do, uncultivated and unimproved as they are.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“You know that the nucleus of a time is not
The poet but the poem, the growth of the mind
Of the world, the heroic effort to live expressed
As victory. The poet does not speak in ruins
Nor stand there making orotund consolations.
He shares the confusions of intelligence.”
—Wallace Stevens (1879–1955)