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Zero (complex analysis)
 
 
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In complex analysis
Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating Function of complex numbers....
, a zero of a holomorphic function
Holomorphic function

Holomorphic functions are the central object of study of complex analysis; they are function defined on an open set of the complex number C with values in C that are complex-differentiable at every point....
 f is a complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 a such that f(a) = 0.

e g is a holomorphic function
Holomorphic function

Holomorphic functions are the central object of study of complex analysis; they are function defined on an open set of the complex number C with values in C that are complex-differentiable at every point....
 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

The multiplicity of a zero aa is also known as the order of vanishing of the function at a.

Existence of zeros
The fundamental theorem of algebra
Fundamental theorem of algebra

In mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex number coefficients has at least one complex root ....
 says that every nonconstant polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
 with complex coefficients has at least one zero in the complex plane
Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
.






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In complex analysis
Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating Function of complex numbers....
, a zero of a holomorphic function
Holomorphic function

Holomorphic functions are the central object of study of complex analysis; they are function defined on an open set of the complex number C with values in C that are complex-differentiable at every point....
 f is a complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 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
Holomorphic function

Holomorphic functions are the central object of study of complex analysis; they are function defined on an open set of the complex number C with values in C that are complex-differentiable at every point....
 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

The multiplicity of a zero aa is also known as the order of vanishing of the function at a.

Existence of zeros


The fundamental theorem of algebra
Fundamental theorem of algebra

In mathematics, the fundamental theorem of algebra states that every non-constant single-variable polynomial with complex number coefficients has at least one complex root ....
 says that every nonconstant polynomial
Polynomial

In mathematics, a polynomial is an expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents....
 with complex coefficients has at least one zero in the complex plane
Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
. This is in contrast to the situation with real
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
 zeros: some polynomial functions with real coefficients have no real zeros. An example is f(x) = x2 + 1.

Properties


An important property of the set of zeros of a holomorphic function (that is not identically zero) 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
Jensen's formula

Jensen's 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....
 and Weierstrass factorization theorem
Weierstrass factorization theorem

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 zero ....
 are results for complex functions which have no counterpart for functions of a real variable.

See also


  • root (mathematics)
    Root (mathematics)

    In mathematics, a root of a complex-valued Function is a member of the Domain of such that vanishes at , that is,In other words, a "root" of a function is a value for that produces a result of zero ....
  • pole (complex analysis)
    Pole (complex analysis)

    In complex analysis, a mathematical discipline, a pole of a meromorphic function is a certain type of mathematical singularity that behaves like the singularity of at ....
  • Hurwitz's theorem
    Hurwitz's theorem

    In mathematics, Hurwitz's theorem is any of at least five different results named after Adolf Hurwitz....
  • filter design
    Filter design

    Filter design is the process of designing a filter , often a linear shift-invariant filter, which satisfies a set of requirements, some of which are contradictory....
  • Nyquist stability criterion
    Nyquist stability criterion

    The Nyquist stability criterion, named after Harry Nyquist, provides a simple test for BIBO stability of a closed-loop control system by examining the open-loop system's Nyquist plot....
     in control theory
    Control theory

    Control theory is an interdisciplinary branch of engineering and mathematics, that deals with the behavior of dynamical systems. The desired output of a system is called the reference....


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