properties of entire functions

1. If  $f:ℂ\to ℂ$  is an entire function and  $z_0\in ℂ$, 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 non-constant 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 non-constant 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.

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