Laurent series

In mathematics, a Laurent series is an infinite series, similar to a power series, but allowing terms of negative degree. Specifically, a Laurent series is an infinite series of the form

··· + a₋₂(x − c)⁻² + a₋₁(x − c)⁻¹ + a₀ + a₁(x − c) + a₂(x − c)² + ···.

The numbers a_k and c are most commonly taken to be complex numbers, although there are other possibilities, as described below. In that context, the letter z is often used in place of x.

Note that terms where a_k = 0 are generally not written, so a specific Laurent series may not appear to go infinitely far to the left or the right. (This is much the same phenomenon as a terminating decimal expansion of a real number, which can actually be thought of as having infinitely many 0 digits to the right.)

A Laurent series with no non-zero terms of negative degree is just a power series. A Laurent series with only a finite number of non-zero terms is called a Laurent polynomial. Finally, a Laurent series with no terms of negative degree and with only a finite number of non-zero terms is a polynomial.

Convergent Laurent series

Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.

e^-1/x² and Laurent approximations: see text for key.

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 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 graph opposite shows e^-1/x² in black and its Laurent approximations

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.

Suppose ∑_−∞<n<∞ a_n(z − c)ⁿ is a given Laurent series with complex coefficients a_n and a complex center c. Then there exists a unique inner radius r and outer radius R such that:

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:

r = lim sup_n→∞ |a_−n|^1/n;

1/R = lim sup_n→∞ |a_n|^1/n.

We take R to be infinite when this latter lim sup is zero.

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/(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 expansion about z = 0 (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 z = 0:

One is defined on the disc where |z| < 1; it's the same as the Taylor series.
One is defined on the annulus where 1 < |z| < 2, caught between the two singularities.
One is defined on the infinite annulus where 2 < |z| < ∞.

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. Then for n in Z, the coefficient a_n of the Laurent series is given by the path integral

a_n = 1/(2πi) ∫_γ f(z)/(z−c)ⁿ⁺¹ dz.

In practice however, 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 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.

For an example of this, consider

f(z) = exp(z)/z + exp(1/z).

This function is holomorphic everwhere except at z = 0. To determine the Laurent expansion about c = 0, we use our knowledge of the Taylor series of the exponential function:

f(z) = ... + (1/3!)z⁻³ + (1/2!)z⁻² + 2z⁻¹ + 2 + (1/2!)z + (1/3!)z² + (1/4!)z³ + ...

and we find that the residue is 2.

Formal Laurent series

Formal Laurent series are Laurent series that are used without regard for their convergence. The coefficients a_k may then be taken from any commutative ring K. In this context, one only considers Laurent series where all but finitely many of the negative-degree coefficients are zero. Furthermore, the center c is taken to be zero.

Two such formal Laurent series are equal if and only if their coefficient sequences are equal. The set of all formal Laurent series in the variable x over the coefficient ring K is denoted by K((x)). Two such formal Laurent series may be added by adding the coefficients, and because of the finiteness of the negative-degree coefficients, they may also be multiplied using convolution of the coefficient sequences. With these two operations, K((x)) becomes a commutative ring.

If K is a field, then the formal power series over K form an integral domain K[[x]]. The field of quotients of this integral domain can be identified with K((x)).