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| A complex number ''a'' is a '''simple zero''' of ''f'', or a '''zero of multiplicity 1''' of ''f'', if ''f'' can be written as | 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>f(z)=(z-a)g(z)\,</math> | ||
| where ''g'' is a holomorphic function ''g'' such that ''g''(''a'') is not zero. | where ''g'' is a holomorphic function ''g'' such that ''g''(''a'') is not zero. | ||
Revision as of 23:22, 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
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
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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