Misplaced Pages

This is an old revision of this page, as edited by Goplat (talk | contribs) at 18:33, 22 November 2004 ({{math-stub}}). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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

${\displaystyle f(z)=(z-a)g(z).}$

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

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

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

This mathematics-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Mathematics stubs
• Complex analysis