Entire Function - Properties

Properties

Every entire function can be represented as a power series that converges everywhere in the complex plane, hence uniformly on compact sets. The Weierstrass factorization theorem asserts that any entire function can be represented by a product involving its zeroes.

The entire functions on the complex plane form an integral domain (in fact a Prüfer domain).

Liouville's theorem states that any bounded entire function must be constant. Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra.

As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere (complex plane and the point at infinity) is constant. Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function. Specifically, by the Casorati–Weierstrass theorem, for any transcendental entire function f and any complex w there is a sequence (z_m)_m∈N with and .

Picard's little theorem is a much stronger result: any non-constant entire function takes on every complex number as value, possibly with a single exception. The latter exception is illustrated by the exponential function, which never takes on the value 0.

Liouville's theorem is a special case of the following statement:

Theorem: Assume M, R are positive constants and that n is a non-negative integer. An entire function f satisfying the inequality for all z with, is necessarily a polynomial, of degree at most n. Similarly, an entire function f satisfying the inequality for all z with, is necessarily a polynomial, of degree at least n.