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/or growth:
“Where mass opinion dominates the government, there is a morbid derangement of the true functions of power. The derangement brings about the enfeeblement, verging on paralysis, of the capacity to govern. This breakdown in the constitutional order is the cause of the precipitate and catastrophic decline of Western society. It may, if it cannot be arrested and reversed, bring about the fall of the West.”
—Walter Lippmann (1889–1974)
“There are enough fagots and waste wood of all kinds in the forests of most of our towns to support many fires, but which at present warm none, and, some think, hinder the growth of the young wood.”
—Henry David Thoreau (1817–1862)