Zero (complex analysis)
Encyclopedia
In complex analysis
, a zero of a holomorphic function
f is a complex number
a such that f(a) = 0.
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
The multiplicity of a zero a is also known as the order of vanishing of the function at a.
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. An example is f(x) = x2 + 1.
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.
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, a zero of a holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
f is a complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
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 aswhere g is a holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
g such that g(a) is not zero.
Generally, the multiplicity
Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial equation has a root at a given point....
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 a is also known as the order of vanishing of the function at a.
Existence of zeros
The fundamental theorem of algebraFundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root...
says that every nonconstant polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
with complex coefficients has at least one zero in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
. This is in contrast to the situation with real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
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 of one variable (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
In the mathematical field known as complex analysis, Jensen's formula, named after Johan Jensen, relates the average magnitude of an analytic function on a circle with the magnitudes of its zeros inside the circle...
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 zeroes...
are results for complex functions which have no counterpart for functions of a real variable.
See also
- Root of a function
- pole (complex analysis)
- Hurwitz's theorem (complex analysis)Hurwitz's theorem (complex analysis)In complex analysis, a field within mathematics, Hurwitz's theorem, named after Adolf Hurwitz, roughly states that, under certain conditions, if a sequence of holomorphic functions converges uniformly to a holomorphic function on compact sets, then after a while those functions and the limit...
- Rouché's theoremRouché's theoremRouché's theorem, named after , states that if the complex-valued functions f and g are holomorphic inside and on some closed contour K, with |g| ...
- filter designFilter designFilter design is the process of designing a filter , often a linear shift-invariant filter, that satisfies a set of requirements, some of which are contradictory...
- Nyquist stability criterionNyquist stability criterionWhen designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that overcompensates -- if the car drifts in one direction, the control system...
in control theoryControl theoryControl 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...