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:
“Put shortly, these are the two views, then. One, that man is intrinsically good, spoilt by circumstance; and the other that he is intrinsically limited, but disciplined by order and tradition to something fairly decent. To the one party man’s nature is like a well, to the other like a bucket. The view which regards him like a well, a reservoir full of possibilities, I call the romantic; the one which regards him as a very finite and fixed creature, I call the classical.”
—Thomas Ernest Hulme (1883–1917)
“I’ve never been afraid to step out and to reach out and to move out in order to make things happen.”
—Victoria Gray, African American civil rights activist. As quoted in This Little Light of Mine, ch. 3, by Hay Mills (1993)
“The idealists dream and the dream is told, and the practical men listen and ponder and bring back the truth and apply it to human life, and progress and growth and higher human ideals come into being and so the world moves ever on.”
—Anna Howard Shaw (1847–1919)