Entire function

In complex analysis, an entire function is a function that is analytic on the whole complex plane.

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 is said to be bounded if there exists a constant such that for all the bound holds. Liouville's theorem yields a simple proof of the fundamental theorem of algebra: if were a polynomial with no roots in the complex plane, then one can prove that would be a bounded entire function, and thus constant.