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.