PlanetMath: properties of entire functions

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properties of entire functions

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If $f\!:\mathbb{C}\to\mathbb{C}$ is an entire function and $z_0\in\mathbb{C}$ , then has the Taylor series expansion
$\displaystyle f(z) = a_0\!+\!a_1(z\!-\!z_0)\!+\!a_2(z\!-\!z_0)^2\!+\cdots$
which is valid in the whole complex plane.
If, conversely, such a power series converges for every complex value , then the sum of the series is an entire function.
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 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 . Examples are complex sine and cosine, complex exponential function, sine integral, error function.
A consequence of Liouville's theorem: If is a non-constant entire function and if and are two arbitrarily great positive numbers, then there exist such points that
$\displaystyle \vert z\vert > R \,\,\,\mathrm{and}\,\,\, \vert f(z)\vert > M.$
This means that the non-constant entire functions are unbounded.
The ring of all entire functions is a Prüfer domain.