Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued 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.
A transcendental entire function is an entire function that is not a polynomial (see transcendental function).
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[edit] 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. Neither the natural logarithm nor the square root functions can be continued analytically to an entire function.
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 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 (zm)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, except possibly one. The latter exception is illustrated by the exponential function, which never takes on the value 0.
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 least n.
[edit] 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 
one can also define the type:
In other words, the order of f(z) is the infimum of all m such that 
as 
. The order need not be finite.
Entire functions may grow as fast as any increasing function: for any increasing function 
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.
[edit] Other examples
J. E. Littlewood chose the Weierstrass sigma function as a 'typical' entire function in one of his books. Other examples include the Fresnel integrals, the Jacobi theta function, and the reciprocal Gamma function. The exponential function and the error function are special cases of the Mittag-Leffler function.
[edit] See also
[edit] Notes
- ^ The converse is also true as for any polynomial
of degree n the inequality 
holds for any |z| ≥ 1.
[edit] References
- Ralph P. Boas (1954). Entire Functions. Academic Press. OCLC 847696.
