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Entire Function

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An entire function is a complex function that is analytic at every finite point of the complex plane C. Entire functions are sometimes also called "integral" (Knopp 1996, p. 112).

Any polynomial a_nz^n+a_(n-1)z^(n-1)+...+a_0 is entire.

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

functionsymbol
Airy functionsAi(z), Bi(z)
Airy function derivativesAi^'(z), Bi^'(z)
Anger functionJ_n(z)
Barnes G-functionG(z)
beibei_n(z)
berber_n(z)
Bessel function of the first kindJ_n(z)
Beurling's functionB(z)
cosinecosz
coversinecovers(z)
Dawson's integralF(z)
erferf(z)
erfcerfc(z)
erfierfi(z)
exponential functione^z=exp(z)
Fresnel integralsC(z), S(z)
gamma function reciprocal1/Gamma(z)
generalized hypergeometric function_pF_q(a_1,...,a_p;b_1,...,b_q;z)
haversinehav(z)
hyperbolic cosinecoshz
hyperbolic sinesinhz
Jacobi theta functionstheta_n(z,q)
Jacobi theta function derivativestheta_n^'(z,q)
Mittag-Leffler functionE_alpha(z)
modified Struve functionL_n(z)
Neville theta functionstheta_c(u), theta_d(u), theta_n(u), theta_s(u)
ShiShi(z)
sinesinz
sine integralSi(z)
spherical Bessel function of the first kindj_n(z)
Struve functionH_n(z)
versinevers(z)
Weber functionsE_n(z)
Wright functionphi(rho,beta;z)
xi-functionxi(z)

The parameters in these examples are held fixed. In the Bessel function of the first kind, ber, bei, spherical Bessel function of the first kind, Struve function, and modified Struve function examples, n is a nonnegative integer.

For the generalized hypergeometric function, take p<=q and no lower parameter b_j equal to a nonpositive integer. Well-defined terminating series are also entire because they are polynomials (NIST, §16.2). For the Mittag-Leffler function, take real alpha>0. For the Wright function, take real rho>-1 and any fixed complex number beta (NIST, §10.46).

For the Jacobi theta functions and their derivatives, n=1, 2, 3, or 4 and the fixed nome satisfies 0<|q|<1. For the Neville theta functions, 0<k<1 is a sufficient choice of fixed elliptic modulus (NIST, §§20.1-20.2).

In contrast, the Bessel function of the second kind has a singularity at the origin, nondegenerate Jacobi elliptic functions have poles, and the standard hyperfactorial continuation has a branch cut. These are not entire functions.

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

See also

Analytic Function, Finite Order, Hadamard Factorization Theorem, Holomorphic Function, Liouville's Boundedness Theorem, Meromorphic Function, Weierstrass Product Theorem

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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.National Institute of Standards and Technology. "Definition and Analytic Properties." §16.2 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/16.2.National Institute of Standards and Technology. "Definitions and Periodic Properties." §20.2 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/20.2.National Institute of Standards and Technology. "Generalized and Incomplete Bessel Functions; Mittag-Leffler Function." §10.46 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.46.National Institute of Standards and Technology. "Special Notation." §20.1 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/20.1.

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Entire Function

Cite this as:

Weisstein, Eric W. "Entire Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EntireFunction.html

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