properties of entire functions

  1. 1.

    If  f:  is an entire functionMathworldPlanetmath and  z0, then f(z) has the Taylor seriesMathworldPlanetmath


    which is valid in the whole complex plane.

  2. 2.

    If, conversely, such a power seriesMathworldPlanetmath converges for every complex value z, then the sum of the series ( is an entire function.

  3. 3.

    The entire functions may be divided in two disjoint :

    a) The entire rational functions, i.e. polynomial functions; in their series there is an n0 such that  an=0nn0.

    b) The entire transcendental functions; in their series one has  an0  for infinitely many values of n.  Examples are complex sine and cosine, complex exponential function, sine integralDlmfDlmfDlmfMathworldPlanetmath, error functionDlmfDlmfPlanetmath.

  4. 4.

    A consequence of Liouville’s theorem:  If f is a non-constant entire function and if R and M are two arbitrarily great positive numbers, then there exist such points z that


    This that the non-constant entire functions are unbounded (

  5. 5.

    The sum (, the product ( and the composition of two entire functions are entire functions.

  6. 6.

    The ring of all entire functions is a Prüfer domain.


  • 1 O. Helmer: “Divisibility properties of integral functions”.  – Duke Math. J. 6 (1940), 345–356.
Title properties of entire functions
Canonical name PropertiesOfEntireFunctions
Date of creation 2013-03-22 14:52:09
Last modified on 2013-03-22 14:52:09
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 19
Author pahio (2872)
Entry type Result
Classification msc 30D20
Related topic RationalFunction
Related topic AlgebraicFunction
Related topic BesselsEquation
Defines entire rational function
Defines entire transcendental function