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| In ], a '''zero''' of a ] ''f'' is a ] ''a'' such that ''f''(''a'') = 0. | In ], a '''zero''' of a ] ''f'' is a ] ''a'' such that ''f''(''a'') = 0. (See also ].) | ||
| ==Multiplicity of a zero== | ==Multiplicity of a zero== | ||
Revision as of 22:11, 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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