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In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...
Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...
Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...
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 This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ...
Existence of zeros 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 zeros: some polynomial functions with real coefficients have no real zeros (but since real numbers are complex numbers, they still have complex zeros). An example is f(x) = x2 + 1. In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree ≥ has some complex root. ...
In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Properties An important property of the set of zeros of a holomorphic function is that the zeros are isolated. In other words, for any zero of a holomorphic function , there is a small disc around the zero which contains no other zeros. There are also some theorems in complex analysis which show the connections between the zeros of a holomorphic (or meromorphic) function and other properties of the function. In particular Jensen's formula and Weierstrass factorization theorem are results for complex functions which have no counterpart for functions of a real variable. Jensens formula in complex analysis relates the behaviour of an analytic function on a circle with the moduli of the zeros inside the circle, and is important in the study of entire functions. ...
In mathematics, the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving their zeroes. ...
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