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)
“Don’t order any black things. Rejoice in his memory; and be radiant: leave grief to the children. Wear violet and purple.... Be patient with the poor people who will snivel: they don’t know; and they think they will live for ever, which makes death a division instead of a bond.”
—George Bernard Shaw (1856–1950)
“Every child has an inner timetable for growth—a pattern unique to him. . . . Growth is not steady, forward, upward progression. It is instead a switchback trail; three steps forward, two back, one around the bushes, and a few simply standing, before another forward leap.”
—Dorothy Corkville Briggs (20th century)