Zero (complex analysis)

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In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0.

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 fundamental theorem of algebra 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) = x2 + 1.

See also