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 expansion $$f(z) = a_0\!+\!a_1(z\!-\!z_0)\!+\!a_2(z\!-\!z_0)^2\!+\cdots$$ which is valid in the whole complex plane.
2. If, conversely, such a power series converges for every complex value $z$ , then the sum of the series is an entire function.
3. The entire functions may be divided in two disjoint classes:

   a) The entire rational functions, i.e. polynomial functions; in their series expansion there is an $n_0$ such that $a_n = 0\,\,\forall n\geqq n_0$ .

   b) The entire transcendental functions; in their series expansion 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. 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 means that the non-constant entire functions are unbounded.
5. The sum, the product and the composition of two entire functions are entire functions.
6. The ring of all entire functions is a Prüfer domain.

Bibliography

1

O. HELMER: ``Divisibility properties of integral functions''. - Duke Math. J. 6 (1940), 345-356.