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. 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).

Contents

Properties

Every entire function can be represented as a power series that 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 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 (zm)m∈N with \lim_{m\to\infty} |z_m| = \infty and \lim_{m\to\infty} f(z_m) = w\ .

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: any entire function f satisfying the inequality |f(z)| \le M |z|^n for all z with |z| \ge R, with n a natural number and M and R positive constants, is necessarily a polynomial, of degree at most n.[1] Conversely, any entire function f satisfying the inequality M |z|^n \le |f(z)| for all z with |z| \ge R, with n a natural number and M and R positive constants, is necessarily a polynomial, of degree at least n.

Order and growth

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

\rho=\limsup_{r\rightarrow\infty}\frac{\log(\log\Vert f \Vert_{\infty, B_r}       )}{\log\, r},

where Br is the disk of radius r and  \Vert f \Vert_{\infty,\,B_r} denotes the supremum norm of f(z) on Br. If 0<\rho<\infty, one can also define the type:

\sigma=\limsup_{r\rightarrow\infty}\frac{\log  \Vert f\Vert_{\infty,B_r}} {r^\rho}.

In other words, the order of f(z) is the infimum of all m such that f(z)=O(\exp\left(|z|^m)\right) as z\to\infty. The order need not be finite.

Entire functions may grow as fast as any increasing function: for any increasing function g:[0,\infty)\to\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:

f(z)=\sum_{k=1}^{\infty}\left(\frac{z}{k}\right)^{n_k},

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.

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.

See also

Notes

  1. ^ The converse is also true as for any polynomial \textstyle p(z) = \sum _{k=0}^na_k z^k of degree n the inequality \textstyle |p(z)| \le \left(\sum_{k=0}^n|a_k|\right) |z|^n holds for any |z| ≥ 1.

References

  • Ralph P. Boas (1954). Entire Functions. Academic Press. OCLC 847696. 

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • entire function — Math. a function of a complex variable that has a derivative for all finite values of the variable. * * * …   Universalium

  • entire function — noun Any function of a complex variable that is holomorphic throughout the complex plane …   Wiktionary

  • entire function — Math. a function of a complex variable that has a derivative for all finite values of the variable …   Useful english dictionary

  • ENTIRE — Function Server Architecture (Originalbild aus dem White Paper 1990) ENTIRE ist ein Produkt der Software AG. Entire Function Server Architecture (EFS) ermöglicht es, Anwendungssysteme aus Komponenten aufzubauen, welche als Services verteilt in… …   Deutsch Wikipedia

  • Entire — *In philately, see Cover. *In mathematics, see Entire function. *In animal fancy and animal husbandry, entire indicates that an animal has not been desexed, that is, spayed (female) or neutered (male). *In botany, an entire leaf is a single whole …   Wikipedia

  • Function (mathematics) — f(x) redirects here. For the band, see f(x) (band). Graph of example function, In mathematics, a function associates one quantity, the a …   Wikipedia

  • Gamma function — For the gamma function of ordinals, see Veblen function. The gamma function along part of the real axis In mathematics, the gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function, with its… …   Wikipedia

  • Bessel-Clifford function — In mathematical analysis, the Bessel Clifford function is an entire function of two complex variables which can be used to provide an alternative development of the theory of Bessel functions. If :pi(x) = frac{1}{Pi(x)} = frac{1}{Gamma(x+1)}is… …   Wikipedia

  • Exponential function — The natural exponential function y = ex In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative …   Wikipedia

  • Dirichlet eta function — For the modular form see Dedekind eta function. Dirichlet eta function η(s) in the complex plane. The color of a point s encodes the value of η(s). Strong colors denote values close to zero and hue encodes the value s argumen …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”