# properties of entire functions

1. 1.

If  $f\!:\mathbb{C}\to\mathbb{C}$  is an entire function and  $z_{0}\in\mathbb{C}$, then $f(z)$ has the Taylor series

 $f(z)=a_{0}\!+\!a_{1}(z\!-\!z_{0})\!+\!a_{2}(z\!-\!z_{0})^{2}\!+\cdots$

which is valid in the whole complex plane.

2. 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. 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  $a_{n}\neq 0$  for infinitely many values of $n$.  Examples are complex sine and cosine, complex exponential function, sine integral, error function.

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

 $|z|>R\,\,\,\mathrm{and}\,\,\,|f(z)|>M.$

This that the non-constant entire functions are unbounded (http://planetmath.org/BoundedFunction).

5. 5.

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

6. 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 PropertiesOfEntireFunctions 2013-03-22 14:52:09 2013-03-22 14:52:09 pahio (2872) pahio (2872) 19 pahio (2872) Result msc 30D20 RationalFunction AlgebraicFunction BesselsEquation entire rational function entire transcendental function