nLab
Laurent series

Contents

Idea

Laurent series generalize power series by allowing both positive and negative powers. In particular, Laurent series with complex coefficients generalize Taylor series of analytic functions to meromorphic functions. A Laurent series for a meromorphic function f(z)f(z) at finite zz\in\mathbb{C} has the form

f(z)= n=k f nz n, f(z) = \sum_{n=k}^{\infty}f_n z^n,

where kk is merely constrained to be finite and is often negative.

Or, in some contexts one wants to take k=k = -\infty. Here such a formal sum (with powers extending infinitely in both directions) is suggestive notation for an element belonging to the dual nRz n\prod_{n \in \mathbb{Z}} R \cdot z^n of a ring nRz n\oplus_{n \in \mathbb{Z}} R \cdot z^n (see Laurent polynomials below). In such contexts, Laurent series can be likened to distributions, i.e., functionals on the algebra of functions R\mathbb{Z} \to R with compact support.

Definition

Definition

A Laurent series in one variable zz over a commutative unital ring kk is a doubly infinite series

f(z)= n= f nz n, f(z) = \sum_{n=-\infty}^{\infty} f_n z^n ,

where f nkf_n\in k. Equivalently: a Laurent series is a function k:nf n\mathbb{Z} \to k: n \mapsto f_n. The kk-module of Laurent series is denoted k[[z,z 1]]k[ [z, z^{-1}] ].

A Laurent polynomial is a Laurent series for which all but finitely many f nf_n are zero. Laurent polynomials form a ring which may be described as k[z,z 1]k[z, z^{-1}] or abstractly as k[x,y]/(xy1)k[x, y]/(x y - 1). Observe that Laurent series in the generality discussed here do not analogously form a ring: the obvious definition of the coefficients of the product h(z)=f(z)g(z)h(z) = f(z)g(z) of two Laurent series f(z),g(z)f(z), g(z),

h n= kf kg nk,h_n = \sum_{k \in \mathbb{Z}} f_k g_{n-k},

doesn’t make sense in general (although it could sometimes make sense in topological contexts where some such infinite sums can converge, as in the case k=k = \mathbb{C}).

Remark

The kk-vector space of Laurent series does however form a module over the ring of Laurent polynomials, i.e., if f(z)k[z,z 1]f(z) \in k[z, z^{-1}] and g(z)g(z) is a Laurent series, then the product h(z)=f(z)g(z)h(z) = f(z)g(z) as defined above always makes sense.

In general, questions of convergence are treated as separate issues. In complex analysis, the Laurent series na nz n\sum_{n \in \mathbb{Z}} a_n z^n describes a meromorphic function in a neighborhood around the point z=0z = 0 (possibly with a pole there) if all but finitely many negatively indexed terms are zero. Similarly, series of the form n=N a n(za) n\sum_{n = -N}^{\infty} a_n (z-a)^n describe meromorphic functions in a neighborhood of z=az=a with poles of order at most NN.

On the other hand, in algebra one often hears of the ring of formal Laurent series. Here, the presence of the word “ring” signifies that we are restricting the coefficients so that multiplication makes sense. Thus,

Definition

The ring of formal Laurent series over a commutative ring AA in an indeterminate xx consists of Laurent series nf nz n\sum_{n \in \mathbb{Z}} f_n z^n, with f nAf_n \in A but where all but finitely many f nf_n for n<0n \lt 0 vanish.

Multiplication defined as above clearly makes sense. If AA is a field kk, then this ring is usually denoted k((x))k((x)) and is in fact a field; indeed it is the field of fractions of the ring k[[x]]k[ [x] ] of formal power series, where k[[x]]k[ [x] ] is often viewed as a discrete valuation ring.

Remark

It would perhaps be clearer if we used the term “restricted Laurent series” to cover the Laurent series considered in Definition 2, and let “Laurent series” be the term that covers doubly infinite series.

Properties

Laurent series as distributions

Another point of view on Laurent series is given in the following alternative definition.

Definition

A Laurent series over kk is a kk-linear functional

ϕ:k[z,z 1]k\phi: k[z, z^{-1}] \to k

on the algebra of Laurent polynomials.

The associated series is nϕ(z n)z n\sum_{n \in \mathbb{Z}} \phi(z^n)z^n. But here we emphasize the point of view that Laurent series have a distribution-like character, with Laurent polynomials being considered a space of functions k\mathbb{Z} \to k with compact (finite) support, via the evident inclusion

k[z,z 1] nkz n nkz n=k .k[z, z^{-1}] \cong \oplus_{n \in \mathbb{Z}} k \cdot z^n \hookrightarrow \prod_{n \in \mathbb{Z}} k \cdot z^n = k^{\mathbb{Z}}.

Of particular interest as a Laurent series is the formal Dirac distribution,

δ(z)= nz n,\delta(z) = \sum_{n \in \mathbb{Z}} z^n,

which intuitively is like the Fourier transform n= e inx\sum_{n = -\infty}^\infty e^{i n x} of the pointwise multiplicative identity 1\mathbf{1} given by 1(n)=1\mathbf{1}(n) = 1 for all nn. It has the following property:

  • For f(z)f(z) a Laurent polynomial, f(z)δ(z)=f(1)δ(z)f(z)\delta(z) = f(1)\delta(z).

Indeed, notice that δ(z)\delta(z) is the distribution pp(1)p \mapsto p(1) which evaluates a Laurent polynomial at the multiplicative identity z=1z=1.

More generally, for ak *a \in k^\ast an invertible unit, the series δ(az)= na nz z\delta(a z) = \sum_{n \in \mathbb{Z}} a^n z^z satisfies

  • f(z)δ(az)=f(a 1)δ(az)f(z)\delta(a z) = f(a^{-1})\delta(a z).

Just as it doesn’t make sense in general to multiply distributions (at least not without heavy qualifications, as in the theory of Colombeau), we cannot make sense of expressions like δ(z) 2\delta(z)^2. However, one can meaningfully work with derivatives of distributions, so we have for example

δ(z)= nnz n\delta'(z) = \sum_{n \in \mathbb{Z}} n z^n

which has the property

  • f(z)δ(z)=f(1)δ(z)f(1)δ(z)f(z)\delta'(z) = f(1)\delta'(z) - f'(1)\delta(z).

Indeed, there is a formal calculus of the Dirac distribution that plays an important role in the theory of vertex operator algebras. See for example the treatment in Frenkel, Lepowsky, Meurman.

One can also work with convolution products h(z)=(f*g)(z)h(z) = (f \ast g)(z), defined by multiplying coefficients degree-wise:

h(z)= nf ng nz n.h(z) = \sum_{n \in \mathbb{Z}} f_n g_n z^n.

Algebraic closure

Theorem

If KK is algebraically closed and has characteristic 0, then the algebraic closure of the field of (restricted) Laurent series K((x))K((x)) over KK is the field of Puiseux series over KK.

Here “restricted” refers to Remark 2. See Puiseux series for more details on this result.

References

  • Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex Operator Algebras and the Monster, Volume 134 in Pure and Applied Mathematics, Academic Press 1988.

See also

For discussion of products of distributions, see

  • J.F. Colombeau, Multiplication of distributions, Bull. Amer. Math. Soc. (N.S.) Volume 23, Number 2 (1990), 251-268. (web)
Revised on September 20, 2013 21:32:38 by Todd Trimble (67.81.95.215)