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Encyclopedia > Laurent series
A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic.

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 cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass discovered it first in 1841 but did not publish it. Image File history File links This is a lossless scalable vector image. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... As the degree of the Taylor series rises, it approaches the correct function. ... Pierre Alphonse Laurent (July 18, 1813 - September 2, 1854) was a French mathematician best known as the discoverer of the Laurent series, an infinite series which bears his name. ... Year 1843 (MDCCCXLIII) was a common year starting on Sunday (link will display the full calendar) of the Gregorian Calendar (or a common year starting on Friday of the 12-day slower Julian calendar). ... Karl Theodor Wilhelm Weierstrass (WeierstraÃŸ) (October 31, 1815 â€“ February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... 1841 is a common year starting on Friday (link will take you to calendar). ...

The Laurent series for a complex function f(z) about a point c is given by:

$f(z)=sum_{n=-infty}^infty a_n(z-c)^n$

where the an are constants, defined by a line integral which is a generalization of Cauchy's integral formula: This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In mathematics, Cauchys integral formula, named after Augustin Louis Cauchy, is a central statement in complex analysis. ...

$a_n=frac{1}{2pi i} oint_gamma frac{f(z),dz}{(z-c)^{n+1}}.,$

The path of integration γ is counterclockwise around a closed, rectifiable path containing no self-intersections, enclosing c and lying in an annulus A in which f(z) is holomorphic. The expansion for f(z) will be valid anywhere inside this annulus. The annulus is shown in red in the diagram on the right, along with an example of a suitable path of integration labelled γ. In practice, this formula is rarely used because the integrals are difficult to evaluate; instead, one typically pieces together the Laurent series by combining known Taylor expansions. The numbers an and c are most commonly taken to be complex numbers, although there are other possibilities, as described below. In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ... An annulus In mathematics, an annulus (the Latin word for little ring, with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ...

e-1/x² and Laurent approximations: see text for key. As the negative degree of the Laurent series rises, it approaches the correct function.

Consider for instance the function f(x) = e−1/x² with f(0) = 0. As a real function, it is infinitely often differentiable everywhere; as a complex function however it is not differentiable at x = 0. By replacing x by −1/x2 in the power series for the exponential function, we obtain its Laurent series which converges and is equal to f(x) for all complex numbers x except at the singularity x=0. The graph opposite shows e−1/x² in black and its Laurent approximations Laurent series approching e^(-1/x²) (made by me) File links The following pages link to this file: Laurent series Categories: GFDL images ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... The exponential function is one of the most important functions in mathematics. ...

$sum_{j=0}^n(-1)^j,{x^{-2j}over j!}$

for n = 1, 2, 3, 4, 5, 6, 7 and 50. As n → ∞, the approximation becomes exact for all (complex) numbers x except at the singularity x = 0.

More generally, Laurent series can be used to express holomorphic functions defined on an annulus, much as power series are used to express holomorphic functions defined on a disc. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... An annulus In mathematics, an annulus (the Latin word for little ring, with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In geometry, a disk is the region in a plane contained inside of a circle. ...

Suppose

$sum_{n=-infty}^{infty} a_n ( z - c )^n$

is a given Laurent series with complex coefficients an and a complex center c. Then there exists a unique inner radius r and outer radius R such that: In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ...

• The Laurent series converges on the open annulus A := {z : r < |z − c| < R}. 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 on compact sets. Finally, the convergent series defines a holomorphic function f(z) on the open annulus.
• Outside the annulus, the Laurent series diverges. That is, at each point of the exterior of A, 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 f(z) cannot be holomorphically continued to those points.

It is possible that r may be zero or R may be infinite; at the other extreme, it's not necessarily true that r is less than R. These radii can be computed as follows: In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. ... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In topology, the exterior of a subset S of a topological space X is the union of all open sets of X which do not meet S. The exterior of S is denoted by ext S or Se. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of...

$r = limsup_{nrightarrowinfty} |a_{-n}|^{1 over n}$
${1 over R} = limsup_{nrightarrowinfty} |a_n|^{1 over n}$

We take R to be infinite when this latter lim sup is zero. In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting bounds on the sequence. ...

Conversely, if we start with an annulus of the form A = {z : r < |z − c| < R} and a holomorphic function f(z) defined on A, then there always exists a unique Laurent series with center c which converges (at least) on A and represents the function f(z).

As an example, let

$f(z) = {1 over (z-1)(z-2i)}$

This function has singularities at z = 1 and z = 2i, where the denominator of the expression is zero and the expression is therefore undefined. A Taylor series about z = 0 (which yields a power series) will only converge in a disc of radius 1, since it "hits" the singularity at 1. As the degree of the Taylor series rises, it approaches the correct function. ... Circle illustration In classical geometry, a radius (plural: radii) of a circle or sphere is any line segment from its center to its boundary. ...

However, there are three possible Laurent expansions about z = 0, depending on the region z is in.

• One is defined on the disc where |z| < 1; it is the same as the Taylor series,
$f(z) = frac{1+2i}{5} sum_{k=0}^infty left(frac{1}{(2i)^{k+1}}-1right)z^k$.
• Another one is defined on the annulus where 1 < |z| < 2, caught between the two singularities,
$f(z) = frac{1+2i}{5} left(sum_{k=1}^infty frac{1}{z^k} + sum_{k=0}^infty frac{1}{(2i)^{k+1}}z^kright)$.
• The third one is defined on the infinite annulus where 2 < |z| < ∞,
$f(z) = frac{1+2i}{5} sum_{k=1}^infty frac{1-(2i)^{k-1}}{z^k}$.

The case r = 0, i.e. a holomorphic function f(z) which may be undefined at a single point c, is especially important.
The coefficient a−1 of the Laurent expansion of such a function is called the residue of f(z) at the singularity c; it plays a prominent role in the residue theorem. In complex analysis, the residue is a complex number which describes the behavior of line integrals of a meromorphic function around a singularity. ... The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. ...

For an example of this, consider

$f(z) = {e^z over z} + e^{1 over z}.$

This function is holomorphic everywhere except at z = 0. To determine the Laurent expansion about c = 0, we use our knowledge of the Taylor series of the exponential function: The exponential function is one of the most important functions in mathematics. ...

$f(z) = cdots + left ( {1 over 3!} right ) z^{-3} + left ( {1 over 2!} right ) z^{-2} + 2z^{-1} + 2 + left ( {1 over 2!} right ) z + left ( {1 over 3!} right ) z^2 + left ( {1 over 4!} right ) z^3 + cdots$

and we find that the residue is 2.

• Formal Laurent series — Laurent series considered formally, with coefficients from an arbitrary commutative ring, without regard for convergence
• Z-transform -- the special case where the Laurent series is taken about zero has much use in Time Series Analysis.

In mathematics, a Laurent series is an infinite series. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics and signal processing, the Z-transform converts a discrete time domain signal, which is a sequence of real numbers, into a complex frequency domain representation. ...

Results from FactBites:

 Laurent series - definition of Laurent series in Encyclopedia (1045 words) In mathematics, a Laurent series is an infinite series. By plugging -1/x² into the series for the exponential function, we obtain its Laurent series which converges and is equal to f(x) for all complex numbers x except at the singularity x=0. The coefficients of the Laurent series can be determined with an integral formula which generalizes Cauchy's integral formula: Pick any rectifiable path γ in A which is closed (has the same beginning and ending points), does not have any self-intersections, and moves around the annulus counterclockwise.
More results at FactBites »

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