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 ...
, the Laurent series of a complex function
is a representation of that function as a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
which includes terms of negative degree. It may be used to express complex functions in cases where a
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 ...
expansion cannot be applied. The Laurent series was named after and first published by
Pierre Alphonse Laurent
Pierre Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician, engineer, and Military Officer best known for discovering the Laurent series, an expansion of a function into an infinite power series, generalizing the T ...
in 1843.
Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.
[.]
Definition
The Laurent series for a complex function
about a point
is given by
where
and
are constants, with
defined by a
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
that generalizes
Cauchy's integral formula:
The path of integration
is counterclockwise around a
Jordan curve enclosing
and lying in an
annulus
Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to:
Human anatomy
* ''Anulus fibrosus disci intervertebralis'', spinal structure
* Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
in which
is
holomorphic (analytic). The expansion for
will then be valid anywhere inside the annulus. The annulus is shown in red in the figure on the right, along with an example of a suitable path of integration labeled
. If we take
to be a circle
, where
, this just amounts
to computing the complex
Fourier coefficients of the restriction of
to
. The fact that these integrals are unchanged by a deformation of the contour
is an immediate consequence of
Green's theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem.
Theorem
Let be a positively orie ...
.
One may also obtain the Laurent series for a complex function
at
. However, this is the same as when
(see the example below).
In practice, the above integral formula may not offer the most practical method for computing the coefficients
for a given function
; instead, one often pieces together the Laurent series by combining known Taylor expansions. Because the Laurent expansion of a function is
unique whenever
it exists, any expression of this form that equals the given function
in some annulus must actually be the Laurent expansion of
.
Convergent Laurent series
Laurent series with complex coefficients are an important tool 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 ...
, especially to investigate the behavior of functions near
singularities.
Consider for instance the function
with
. As a real function, it is infinitely differentiable everywhere; as a complex function however it is not differentiable at
. By replacing
with
in the
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
for the
exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
, we obtain its Laurent series which converges and is equal to
for all complex numbers
except at the singularity
. The graph opposite shows
in black and its Laurent approximations
for
=
1,
2,
3,
4,
5,
6,
7 and
50. As
, the approximation becomes exact for all (complex) numbers
except at the singularity
.
More generally, Laurent series can be used to express
holomorphic functions defined on an
annulus
Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to:
Human anatomy
* ''Anulus fibrosus disci intervertebralis'', spinal structure
* Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
, much as
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
are used to express holomorphic functions defined on a
disc.
Suppose
is a given Laurent series with complex coefficients
and a complex center
. Then there exists a
unique inner radius
and outer radius
such that:
* The Laurent series converges on the open annulus
. To say that the Laurent series converges, we mean that both the positive degree power series and the negative degree power series converge. Furthermore, this convergence will be
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
on
compact sets. Finally, the convergent series defines a
holomorphic function on the open annulus.
* Outside the annulus, the Laurent series diverges. That is, at each point of the
exterior of
, the positive degree power series or the negative degree power series diverges.
* On the
boundary of the annulus, one cannot make a general statement, except to say that there is at least one point on the inner boundary and one point on the outer boundary such that
cannot be holomorphically continued to those points.
It is possible that
may be zero or
may be infinite; at the other extreme, it's not necessarily true that
is less than
.
These radii can be computed as follows:
We take
to be infinite when this latter
lim sup
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
is zero.
Conversely, if we start with an annulus of the form
and a holomorphic function
defined on
, then there always exists a unique Laurent series with center
which converges (at least) on
and represents the function
.
As an example, consider the following rational function, along with its
partial fraction
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction a ...
expansion:
This function has singularities at
and
, where the denominator of the expression is zero and the expression is therefore undefined.
A
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 ...
about
(which yields a power series) will only converge in a disc of
radius
In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...
1, since it "hits" the singularity at 1.
However, there are three possible Laurent expansions about 0, depending on the radius of
:
* One series is defined on the inner disc where ; it is the same as the Taylor series,
This follows from the partial fraction form of the function, along with the formula for the sum of a
geometric series,
for
.
* The second series is defined on the middle annulus where