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 $f\$ is said to be bounded if there exists a constant $M\$ such that for all $z\in \mathbb {C} \$ the bound $f(z)<M\$ holds. Liouville's theorem yields a simple proof of the fundamental theorem of algebra: if $p(z)\$ were a polynomial with no roots in the complex plane, then one can prove that $1/p(z)\$ would be a bounded entire function, and thus constant.