Misplaced Pages

Zero (complex analysis)

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.

This is an old revision of this page, as edited by Michael Hardy (talk | contribs) at 22:12, 7 January 2005 (Some browsers fail to render the simplest cases of TeX as TeX without these otherwise seemingly epiphenomenal spacing characters.). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Revision as of 22:12, 7 January 2005 by Michael Hardy (talk | contribs) (Some browsers fail to render the simplest cases of TeX as TeX without these otherwise seemingly epiphenomenal spacing characters.)(diff) ← Previous revision | Latest revision (diff) | Newer revision → (diff)

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 a is not a zero of the holomorophic function g such that

f ( z ) = ( z a ) g ( z ) . {\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

f ( z ) = ( z a ) n g ( z )   and   g ( a ) 0. {\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 nonconstant 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

Stub icon

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

Categories: