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In ], a '''zero''' of a ] ''f'' is a ] ''a'' such that ''f''(''a'') = 0. (See also ].) In ], a '''zero''' of a ] ''f'' is a ] ''a'' such that ''f''(''a'') = 0. See also ].


==Multiplicity of a zero== ==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 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> :<math>f(z)=(z-a)g(z).\,</math>
where ''g'' is a holomorphic function ''g'' such that ''g''(''a'') is not zero.


Generally, the ''']''' of the zero of ''f'' at ''a'' is the positive integer ''n'' for which there is a holomorphic function ''g'' such that Generally, the ''']''' of the zero of ''f'' at ''a'' is the positive integer ''n'' for which there is a holomorphic function ''g'' such that
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==Existence of zeroes== ==Existence of zeroes==


The so-called ] (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 ] 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''<sup>2</sub> + 1. The so-called ] (something of a misnomer) says that every nonconstant ] with complex coefficients has at least one zero in the complex plane. This is in contrast to the situation with ] 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''<sup>2</sub> + 1.


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Revision as of 23:21, 18 January 2005

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

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

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

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 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

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