Zero (complex analysis)

In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0.

Multiplicity of a zero

A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if a is not a zero of the holomorophic function g such that

$f(z)=(z-a)g(z).$

Generally, the multiplicity of the zero of f at a is the positive integer n for which there is a holomorphic function g such that

$f(z)=(z-a)^{n}g(z)\ {\mbox{and}}\ g(a)\neq 0.$

Existence of zeroes

The so-called fundamental theorem of algebra (something of a misnomer) says that every polynomial function with complex coefficients has at least one zero in the complex plane. This is in contrast to the situation with real zeroes: some polynomial functions with real coefficients have no real zeroes (but since real numbers are complex numbers, they still have complex zeroes). An example is f(x) = x^{2 + 1.}