Zero (complex analysis) - Definition and Overview

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

Multiplicity of a zero

A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written as

<math>f(z)=(z-a)g(z)\,<math>

where g is a holomorphic function g such that g(a) is not zero.

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

<math>f(z)=(z-a)^ng(z)\ \mbox{and}\ g(a)\neq 0.\,<math>

Existence of zeroes

The so-called fundamental theorem of algebra (something of a misnomer) says that every nonconstant polynomial 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</sub> + 1.}