In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
(a branch of mathematics), a pole is a certain type of
singularity of a
complex-valued function of a
complex variable. In some sense, it is the simplest type of singularity. Technically, a point is a pole of a function if it is a
zero of the function and is
holomorphic in some
neighbourhood of (that is,
complex differentiable
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
in a neighbourhood of ).
A function is
meromorphic in an
open set if for every point of there is a neighborhood of in which either or is holomorphic.
If is meromorphic in , then a zero of is a pole of , and a pole of is a zero of . This induces a duality between ''zeros'' and ''poles'', that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
plus the
point at infinity, then the sum of the
multiplicities
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 has a root at a given point is the multiplicity of that root.
The notion of multip ...
of its poles equals the sum of the multiplicities of its zeros.
Definitions
A
function of a complex variable
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
is
holomorphic in an
open domain
Question answering (QA) is a computer science discipline within the fields of information retrieval and natural language processing (NLP), which is concerned with building systems that automatically answer questions posed by humans in a natural la ...
if it is
differentiable with respect to at every point of . Equivalently, it is holomorphic if it is
analytic, that is, if its
Taylor series exists at every point of , and converges to the function in some
neighbourhood of the point. A function is
meromorphic in if every point of has a neighbourhood such that either or is holomorphic in it.
A
zero of a meromorphic function is a complex number such that . A pole of is a zero of .
If is a function that is meromorphic in a neighbourhood of a point
of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, then there exists an integer such that
:
is holomorphic and nonzero in a neighbourhood of
(this is a consequence of the analytic property).
If , then
is a ''pole'' of order (or multiplicity) of . If , then
is a zero of order
of . ''Simple zero'' and ''simple pole'' are terms used for zeroes and poles of order
''Degree'' is sometimes used synonymously to order.
This characterization of zeros and poles implies that zeros and poles are
isolated
Isolation is the near or complete lack of social contact by an individual.
Isolation or isolated may also refer to:
Sociology and psychology
*Isolation (health care), various measures taken to prevent contagious diseases from being spread
**Is ...
, that is, every zero or pole has a neighbourhood that does not contain any other zero and pole.
Because of the ''order'' of zeros and poles being defined as a non-negative number and the symmetry between them, it is often useful to consider a pole of order as a zero of order and a zero of order as a pole of order . In this case a point that is neither a pole nor a zero is viewed as a pole (or zero) of order 0.
A meromorphic function may have infinitely many zeros and poles. This is the case for the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
(see the image in the infobox), which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. The
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
is also meromorphic in the whole complex plane, with a single pole of order 1 at . Its zeros in the left halfplane are all the negative even integers, and the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
is the conjecture that all other zeros are along .
In a neighbourhood of a point
a nonzero meromorphic function is the sum of a
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
with at most finite ''principal part'' (the terms with negative index values):
:
where is an integer, and
Again, if (the sum starts with
, the principal part has terms), one has a pole of order , and if (the sum starts with
, there is no principal part), one has a zero of order
.
At infinity
A function
is ''meromorphic at infinity'' if it is meromorphic in some neighbourhood of infinity (that is outside some
disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
), and there is an integer such that
:
exists and is a nonzero complex number.
In this case, the
point at infinity is a pole of order if , and a zero of order
if .
For example, a
polynomial of degree has a pole of degree at infinity.
The
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
extended by a point at infinity is called the
Riemann sphere.
If is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros.
Every
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum of the degrees of the numerator and the denominator.
Examples
* The function
::
: is meromorphic on the whole Riemann sphere. It has a pole of order 1 or simple pole at
and a simple zero at infinity.
* The function
::
: is meromorphic on the whole Riemann sphere. It has a pole of order 2 at
and a pole of order 3 at
. It has a simple zero at
and a quadruple zero at infinity.
* The function
::
: is meromorphic in the whole complex plane, but not at infinity. It has poles of order 1 at
. This can be seen by writing the
Taylor series of
around the origin.
* The function
::
: has a single pole at infinity of order 1, and a single zero at the origin.
All above examples except for the third are
rational functions. For a general discussion of zeros and poles of such functions, see .
Function on a curve
The concept of zeros and poles extends naturally to functions on a ''complex curve'', that is
complex analytic manifold of dimension one (over the complex numbers). The simplest examples of such curves are the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
and the
Riemann surface. This extension is done by transferring structures and properties through
chart
A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabu ...
s, which are analytic
isomorphisms.
More precisely, let be a function from a complex curve to the complex numbers. This function is holomorphic (resp. meromorphic) in a neighbourhood of a point of if there is a chart
such that
is holomorphic (resp. meromorphic) in a neighbourhood of
Then, is a pole or a zero of order if the same is true for
If the curve is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, and the function is meromorphic on the whole curve, then the number of zeros and poles is finite, and the sum of the orders of the poles equals the sum of the orders of the zeros. This is one of the basic facts that are involved in
Riemann–Roch theorem.
See also
*
*
Filter design
*
Filter (signal processing)
In signal processing, a filter is a device or process that removes some unwanted components or features from a signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspe ...
*
Gauss–Lucas theorem
*
Hurwitz's theorem (complex analysis)
*
Marden's theorem
*
Nyquist stability criterion
*
Pole–zero plot
In mathematics, signal processing and control theory, a pole–zero plot is a graphical representation of a rational transfer function in the complex plane which helps to convey certain properties of the system such as:
* Stability
* Causal syste ...
*
Residue (complex analysis)
*
Rouché's theorem
*
Sendov's conjecture
References
*
*
*
External links
* {{MathWorld , urlname= Pole , title= Pole
Complex analysis