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| 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 ''a'' is ''not'' a zero of the holomorophic function ''g'' such that | ||
| :<math>f(z)=(z-a)g(z).</math> | :<math>f(z)=(z-a)g(z).\,</math> | ||
| 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 | ||
| :<math>f(z)=(z-a)^ng(z)\ \mbox{and}\ g(a)\neq 0.</math> | :<math>f(z)=(z-a)^ng(z)\ \mbox{and}\ g(a)\neq 0.\,</math> | ||
| ==Existence of zeroes== | ==Existence of zeroes== | ||
Revision as of 22:12, 7 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 a is not a zero of the holomorophic function g such that
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
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
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