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:
“In the order of literature, as in others, there is no act that is not the coronation of an infinite series of causes and the source of an infinite series of effects.”
—Jorge Luis Borges (18991986)
“We already have the statistics for the future: the growth percentages of pollution, overpopulation, desertification. The future is already in place.”
—Günther Grass (b. 1927)