Laurent Series

Introduction

Taylor series represent analytic functions as power series with non-negative powers. But what about functions with singularities? Laurent series extend Taylor series by including negative powers, allowing us to represent functions with poles and other isolated singularities.

Laurent series are essential tools in complex analysis. They classify singularities, enable residue calculations, and provide representations of functions in annular regions where Taylor series fail.

Named after Pierre Alphonse Laurent (1843), these series bridge local and global properties of complex functions.

Definition

The Laurent series of ƒ(z) centered at (z_0) is:

ƒ(z)=(∑_n=−∞^∞)((a_n))*(z−(z_0))n=⋯+(a_−2)/((z−(z_0))2)+(a_−1)/(z−(z_0))+(a_0)+(a_1)*(z−(z_0))+(a_2)*(z−(z_0))2+⋯

The series has two parts:

• Analytic part (principal part): non-negative powers n≥0

• Principal part (singular part): negative powers n<0

Region of Convergence

A Laurent series converges in an annulus:

r<|z−(z_0)|<R

where r≥0 is the inner radius and R≤∞ is the outer radius. The function may have singularities inside the inner circle or outside the outer circle.

A Taylor series is the special case r=0 (no negative powers needed).

Computing Coefficients

The coefficients are given by contour integrals:

(a_n)=1/(2*π*i)*(∮_C)(ƒ(z)/((z−(z_0))(n+1))*d(z))

where C is any positively oriented simple closed contour in the annulus encircling (z_0).

In practice, we often find Laurent series by algebraic manipulation, partial fractions, or known expansions.

Classification of Singularities

The Laurent series at an isolated singularity (z_0) classifies it:

Removable Singularity

No negative powers ((a_n)=0 for all n<0). The function can be extended to be analytic at (z_0).

Example: ƒ(z)=sin(z)/z at z=0. Laurent series:1−z2/6+z4/120−…

Pole of Order m

Finitely many negative powers, with (a_−m)≠0 being the most negative.

Example: ƒ(z)=1/(z2) at z=0 is a pole of order 2.

A simple pole has order 1.

Essential Singularity

Infinitely many negative powers.

Example: ƒ(z)=e(1/z) at z=0. Laurent series: 1+1/z+1/(2!z2)+1/(3!z3)+…

Near an essential singularity, ƒ takes every complex value (except possibly one) infinitely often (Picard's theorem).

The Residue

The coefficient (a_−1) is called the residue of ƒ at (z_0):

Res(ƒ,(z_0))=(a_−1)=1/(2*π*i)*(∮_C)(ƒ(z)*d(z))

The residue is the key to evaluating contour integrals via the Residue Theorem.

Techniques for Finding Laurent Series

1. Partial fractions: Split rational functions

2. Geometric series: 1/(1−w)=1+w+w2+… for |w|<1

3. Known Taylor series: Substitute and adjust

4. Differentiation/integration: Derive from simpler series

Connection to Residue Theorem

The Residue Theorem states:

(∮_C)(ƒ(z)*d(z))=2*π*i*(∑_k^)(Res(ƒ,(z_k)))

where the sum is over all singularities inside C. The residue is exactly the (a_−1) coefficient in the Laurent series.

Summary

Laurent series extend Taylor series by including negative powers, allowing representation of functions with isolated singularities. The series converges in an annulus r<|z−(z_0)|<R.

The principal part (negative powers) classifies singularities: none = removable, finite = pole, infinite = essential. The coefficient (a_−1) is the residue, crucial for contour integration.

Laurent series are computed using partial fractions, geometric series, and known expansions. Different annular regions around the same point can yield different Laurent series.