Entire Function

If a complex function is analytic at all finite points of the complex plane , then it is said to be entire, sometimes also called "integral" (Knopp 1996, p. 112).

Any polynomial  is entire.

Examples of specific entire functions are given in the following table.

function	symbol
Airy functions	,
Airy function derivatives	,
Anger function
Barnes G-function
bei
ber
Bessel function of the first kind
Bessel function of the second kind
Beurling's function
cosine
coversine
Dawsons integral
erf
erfc
erfi
exponential function
Fresnel integrals	,
gamma function reciprocal
generalized hypergeometric function
haversine
hyperbolic cosine
hyperbolic sine
Jacobi elliptic functions	, , , , , , , , , , ,
Jacobi theta functions
Jacobi theta function derivatives
Mittag-Leffler function
modified Struve function
Neville theta functions	, , ,
Shi
sine
sine integral
spherical Bessel function of the first kind
Struve function
versine
Weber functions
Wright function
xi-function

Liouville's boundedness theorem states that a bounded entire function must be a constant function.

REFERENCES:

Knopp, K. "Entire Transcendental Functions." Ch. 9 in Theory of Functions Parts I and II, Two Volumes Bound as One, Part I.New York: Dover, pp. 112-116, 1996.

Krantz, S. G. "Entire Functions and Liouville's Theorem." §3.1.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 31-32, 1999.