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| In ], a '''zero''' of a ] ''f'' is a ] ''a'' such that ''f''(''a'') = 0. (See also ].) | |||
| {{R from merge}} | |||
| ==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 | |||
| :<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 | |||
| :<math>f(z)=(z-a)^ng(z)\ \mbox{and}\ g(a)\neq 0.\,</math> | |||
| ==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. | |||
| {{math-stub}} | |||
| ] | |||
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