In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically
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 ...
, the residue is a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
proportional to the
contour integral of a
meromorphic function along a path enclosing one of its
singularities. (More generally, residues can be calculated for any function
that is
holomorphic except at the discrete points
''k'', even if some of them are
essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the
residue theorem.
Definition
The residue of a
meromorphic function at an
isolated singularity , often denoted
,
,
or
, is the unique value
such that
has an
analytic
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ...
in a
punctured disk
In mathematics, an annulus (plural annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word ''anulus'' or ''annulus'' mea ...
.
Alternatively, residues can be calculated by finding
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 ...
expansions, and one can define the residue as the coefficient ''a''
−1 of a Laurent series.
The definition of a residue can be generalized to arbitrary
Riemann surfaces
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
. Suppose
is a
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
on a Riemann surface. Let
be meromorphic at some point
, so that we may write
in local coordinates as
. Then, the residue of
at
is defined to be the residue of
at the point corresponding to
.
Examples
Residue of a monomial
Computing the residue of a
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expon ...
:
makes most residue computations easy to do. Since path integral computations are
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
invariant, we will let
be the circle with radius
. Then, using the change of coordinates
we find that
:
hence our integral now reads as
:
Application of monomial residue
As an example, consider the
contour integral
:
where ''C'' is some
simple closed curve
In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exteri ...
about 0.
Let us evaluate this integral using a standard convergence result about integration by series. We can substitute the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
for
into the integrand. The integral then becomes
:
Let us bring the 1/''z''
5 factor into the series. The contour integral of the series then writes
:
Since the series converges uniformly on the support of the integration path, we are allowed to exchange integration and summation.
The series of the path integrals then collapses to a much simpler form because of the previous computation. So now the integral around ''C'' of every other term not in the form ''cz''
−1 is zero, and the integral is reduced to
:
The value 1/4! is the ''residue'' of ''e''
''z''/''z''
5 at ''z'' = 0, and is denoted
:
Calculating residues
Suppose a
punctured disk
In mathematics, an annulus (plural annuli or annuluses) is the region between two concentric circles. Informally, it is shaped like a ring or a hardware washer. The word "annulus" is borrowed from the Latin word ''anulus'' or ''annulus'' mea ...
''D'' = in the complex plane is given and ''f'' is a
holomorphic function
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 deriv ...
defined (at least) on ''D''. The residue Res(''f'', ''c'') of ''f'' at ''c'' is the coefficient ''a''
−1 of in the
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 ...
expansion of ''f'' around ''c''. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity.
According to the
residue theorem, we have:
:
where ''γ'' traces out a circle around ''c'' in a counterclockwise manner. We may choose the path ''γ'' to be a circle of radius ''ε'' around ''c'', where ''ε'' is as small as we desire. This may be used for calculation in cases where the integral can be calculated directly, but it is usually the case that residues are used to simplify calculation of integrals, and not the other way around.
Removable singularities
If the function ''f'' can be
continued to a
holomorphic function
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 deriv ...
on the whole disk