In complex analysis, an entire function, also called an integral function, is a complexvalued function that is holomorphic over the whole complex plane. Typical examples of entire functions are the polynomials and the exponential function, and any sums, products and compositions of these, including the error function and the trigonometric functions sine and cosine and their hyperbolic counterparts the hyperbolic sine and hyperbolic cosine functions. Neither the natural logarithm nor the square root functions can be continued analytically to an entire function.
A transcendental entire function is an entire function that is not a polynomial (see transcendental function).
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Properties
Every entire function can be represented as a power series which converges 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 a commutative ring (in fact a Prüfer domain).
Any entire function f satisfying the inequality for all z with , with n a natural number and M and R positive constants, is necessarily a polynomial, of degree at most n.^{[1]}
The special case n = 0 is called Liouville's theorem: 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 which is entire on the whole Riemann sphere (complex plane and the point at infinity) is constant. Thus any nonconstant 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 .
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