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Added brief discussion of Liouville's theorem.
In [[complex analysis]], an '''entire function''' is a [[function]] that is [[analytic]] on the whole [[complex plane]].
In [[complex analysis]], an '''entire function''' is a [[function]] that is [[analytic]] on the whole [[complex plane]].
[[category:Complex analysis]]
[[category:Complex analysis]].
The main result governing the behavior of entire functions is Liouville's theorem, which states that a bounded entire function is constant. Here an entire function <math>f</math> is said to be bounded if there exists a constant <math>M</math> such that for all <math>z \in \mathbb C</math> the bound <math>f(z)<M</math> holds. Liouville's theorem yields a simple proof of the fundamental theorem of algabra: if <math>p(z)</math> were a polynomial with no roots in the complex plane, then one can prove that <math>1/p(z)</math> would be a bounded entire function, and thus constant.