properties of entire functions

1.
If $f:\u2102\to \u2102$ is an entire function^{} and ${z}_{0}\in \u2102$, then $f(z)$ has the Taylor series^{}
$$f(z)={a}_{0}+{a}_{1}(z{z}_{0})+{a}_{2}{(z{z}_{0})}^{2}+\mathrm{\cdots}$$ which is valid in the whole complex plane.

2.
If, conversely, such a power series^{} converges for every complex value $z$, then the sum of the series (http://planetmath.org/SumFunctionOfSeries) is an entire function.

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 ${n}_{0}$ such that ${a}_{n}=0\forall n\geqq {n}_{0}$.
b) The entire transcendental functions; in their series one has ${a}_{n}\ne 0$ for infinitely many values of $n$. Examples are complex sine and cosine, complex exponential function, sine integral^{}, error function^{}.

4.
A consequence of Liouville’s theorem: If $f$ is a nonconstant entire function and if $R$ and $M$ are two arbitrarily great positive numbers, then there exist such points $z$ that
$$z>R\mathrm{and}f(z)>M.$$ This that the nonconstant entire functions are unbounded (http://planetmath.org/BoundedFunction).

5.
The sum (http://planetmath.org/SumOfFunctions), the product (http://planetmath.org/ProductOfFunctions) and the composition of two entire functions are entire functions.

6.
The ring of all entire functions is a Prüfer domain.
References
 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  20130322 14:52:09 
Last modified on  20130322 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 