Entire Function - Order and Growth

Order and Growth

The order (at infinity) of an entire function f(z) is defined using the limit superior as:

where B_r is the disk of radius r and denotes the supremum norm of f(z) on B_r. 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 n_k. 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.