Laurent Series - Convergent Laurent Series

Convergent Laurent Series

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 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 ƒ(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

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 ƒ(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 ƒ(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:

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 ƒ(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 ƒ(z).

As an example, let

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.

However, there are three possible Laurent expansions about 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,

(The technique involves using partial fractions to split the original expression for f(z) into two simpler fractions and then exploiting the fact that 1/(1-z) is the formula for the sum of a geometric series with first term 1 and constant multiplier z.)

• Another one is defined on the annulus where 1 < |z| < 2, caught between the two singularities,

• The third one is defined on the infinite annulus where 2 < |z| < ∞,

(The terms above can be derived through polynomial long division or using the sum of a geometric series trick again, this time using and as the common ratios.)

The case r = 0, i.e. a holomorphic function ƒ(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 ƒ(z) at the singularity c; it plays a prominent role in the residue theorem.

For an example of this, consider

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:

and we find that the residue is 2.