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).

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 (z_m)_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 B_r is the disk of radius r and $\Vert f \Vert_{\infty,\,B_r}$ denotes the supremum norm of f(z) on B_r. 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 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.

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

• Jensen's formula
• Carlson's theorem
• Exponential type
• Paley–Wiener theorem

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.