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:
“Success and failure in our own national economy will hang upon the degree to which we are able to work with races and nations whose social order and whose behavior and attitudes are strange to us.”
—Ruth Benedict (18871948)
“Have you noticed when reading War and Peace the difficulties Tolstoy experienced in forcing morally wounded Bolkonsky to come into geographical and chronological contact with Natasha? It is very painful to watch the way the poor fellow is dragged and pushed and shoved in order to achieve this happy reunion.”
—Vladimir Nabokov (18991977)
“Of all the wastes of human ignorance perhaps the most extravagant and costly to human growth has been the waste of the distinctive powers of womanhood after the child-bearing age.”
—Anna Garlin Spencer (18511931)