Laurent Series

Doubly Infinite Series

A doubly infinite series is a series of the form: And define the convergence of a doubly infinite series by breaking it into two series: and we say that the doubly infinite series converges (absolutely) if both series on the righthand side converge (absolutely).

Warning

is a different notion of convergence.

Laurent Series

A Laurent series centered at is a doubly infinite series of the form:

Proposition

A Laurent series converges normally and absolutely on an annulus: and diverges outside the annulus, as long as the series converges somewhere.

Proof Any Laurent series can be written as follows: The first series converges normally and absolutely if , that is on , and the second series converges absolutely on , so the Laurent series converges absolutely on , letting . Note that and are entirely independent, t is possible that , in which case .

Corollary

The limit of a Laurent series lies in , where is the annulus of convergence.

Laurent Expansion

Let . Then there is a unique Laurent series representation centered at that converges normally and absolutely to . Explicitly, the coefficients are given by: where .

Proof Without loss of generality, assume . Because every Laurent series converges normally and absolutely on its annulus of convergence, it suffices to show that there exists a Laurent series converging pointwise to . Pick some and satisfying . Since , we can apply the Cauchy integral formula: for . For the green term, we can write: where we used normal convergence to exchange the sum and the integral in the last equality. Similarly, we can write the blue term as: Therefore, we have: Notice that by homotopy invariance, the integrals are independent of the choice of and , so for any . Now prove the uniqueness. Suppose converges normally to on . Then we can write: When , since has a primitive on . When , . Therefore, we have: Thus Hence the coefficients are unique.