Laurent series

In mathematics, the Laurent series of a complex function 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 may have discovered it first in a paper written in 1841, but it was not published until after his death.[1]

A Laurent series is defined with respect to a particular point and a path of integration γ. The path of integration must lie in an annulus, indicated here by the red color, inside which is holomorphic (analytic).

DefinitionEdit

The Laurent series for a complex function   about a point   is given by

 
where   and   are constants, with   defined by a line integral that generalizes Cauchy's integral formula:
 

The path of integration   is counterclockwise around a Jordan curve enclosing   and lying in an annulus   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.

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 seriesEdit

 
e−1/x2 and Laurent approximations: see text for key. As the negative degree of the Laurent series rises, it approaches the correct function.
 
e−1/x2 and its Laurent approximations with the negative degree rising. The neighborhood around the zero singularity can never be approximated.

Laurent series with complex coefficients are an important tool in complex analysis, 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 for the exponential function, 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, much as power series 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 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 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 expansion:

 

This function has singularities at   and  , where the denominator of the expression is zero and the expression is therefore undefined. A Taylor series about   (which yields a power series) will only converge in a disc of 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 |z| < 1; 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   is caught between the two singularities:
     
    Here, we use the alternative form of the geometric series summation,
     
    for  .
  • The third series is defined on the infinite outer annulus where  , (which is also the Laurent expansion at  )
     
    This series can be derived using geometric series as before, or by performing polynomial long division of 1 by  , not stopping with a remainder but continuing into   terms; indeed, the "outer" Laurent series of a rational function is analogous to the decimal form of a fraction. (The "inner" Taylor series expansion can be obtained similarly, just reversing the term order in the division algorithm.)

The case  ; i.e., a holomorphic function   which may be undefined at a single point  , is especially important. The coefficient   of the Laurent expansion of such a function is called the residue of   at the singularity  ; it plays a prominent role in the residue theorem. For an example of this, consider

 

This function is holomorphic everywhere except at  .

To determine the Laurent expansion about  , we use our knowledge of the Taylor series of the exponential function:

 

We find that the residue is 2.

One example for expanding about  :

 

UniquenessEdit

Suppose a function   holomorphic on the annulus   has two Laurent series:

 

Multiply both sides by  , where k is an arbitrary integer, and integrate on a path γ inside the annulus,

 

The series converges uniformly on  , where ε is a positive number small enough for γ to be contained in the constricted closed annulus, so the integration and summation can be interchanged. Substituting the identity

 
into the summation yields
 

Hence the Laurent series is unique.

Laurent polynomialsEdit

A Laurent polynomial is a Laurent series in which only finitely many coefficients are non-zero. Laurent polynomials differ from ordinary polynomials in that they may have terms of negative degree.

Principal partEdit

The principal part of a Laurent series is the series of terms with negative degree, that is

 

If the principal part of   is a finite sum, then   has a pole at   of order equal to (negative) the degree of the highest term; on the other hand, if   has an essential singularity at  , the principal part is an infinite sum (meaning it has infinitely many non-zero terms).

If the inner radius of convergence of the Laurent series for   is 0, then   has an essential singularity at   if and only if the principal part is an infinite sum, and has a pole otherwise.

If the inner radius of convergence is positive,   may have infinitely many negative terms but still be regular at  , as in the example above, in which case it is represented by a different Laurent series in a disk about  .

Laurent series with only finitely many negative terms are well-behaved—they are a power series divided by  , and can be analyzed similarly—while Laurent series with infinitely many negative terms have complicated behavior on the inner circle of convergence.

Multiplication and sumEdit

Laurent series cannot in general be multiplied. Algebraically, the expression for the terms of the product may involve infinite sums which need not converge (one cannot take the convolution of integer sequences). Geometrically, the two Laurent series may have non-overlapping annuli of convergence.

Two Laurent series with only finitely many negative terms can be multiplied: algebraically, the sums are all finite; geometrically, these have poles at  , and inner radius of convergence 0, so they both converge on an overlapping annulus.

Thus when defining formal Laurent series, one requires Laurent series with only finitely many negative terms.

Similarly, the sum of two convergent Laurent series need not converge, though it is always defined formally, but the sum of two bounded below Laurent series (or any Laurent series on a punctured disk) has a non-empty annulus of convergence.

Also, for a field  , by the sum and multiplication defined above, formal Laurent series would form a field   which is also the field of fractions of the ring   of formal power series.

See alsoEdit

ReferencesEdit

  1. ^ Rodriguez, Rubi; Kra, Irwin; Gilman, Jane P. (2012), Complex Analysis: In the Spirit of Lipman Bers, Graduate Texts in Mathematics, vol. 245, Springer, p. 12, ISBN 9781441973238.

External linksEdit