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 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 f \ } is said to be bounded if there exists a constant 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 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.