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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/p(z)} would be a bounded entire function, and thus constant.