A **Laurent series** is a way to represent a complex function f(z) as a complex power series with negative powers.

This generalization of the Taylor series has two major advantages:

- The series can include both positive and negative powers,
- It can be expanded around singularities to analyze functions in neighborhoods around those singularities.

Watch the video for a short introduction or read on below:

## Why These Series Are Useful

When these series represent deleted neighborhoods around a singularity, they can be useful for identifying essential discontinuities. Specifically, a singularity is essential if the principal part of the Laurent series has infinitely many nonzero terms (Kramer, n.d.).

A punctured disk (a disc with a pinprick in the center), 0 < |z – z_{0} | < can also be written in terms of a *Laurent series*. Let’s say the series is:

Then we can say that z_{0} is a pole of order p.

## Use of Laurent Series vs Taylor Series

A Taylor series expansion can only express a function as a series with non-negative powers, so the Laurent series becomes very useful when you can’t use a Taylor series. Another way to think of a Laurent series is that—unlike the Taylor series—it allows for the existence of poles.

## Formal Definition

If a function f(z) is analytic on the annulus:

The function can be represented by (Orloff, 2020):

The **principal part** of the series is any term with negative powers of z – z_{0}. This part can include a finite number or terms, or an infinite number of terms (Stephenson & Radmore, 1990).

The Laurent series is a natural generalization of the Taylor series when the expansion center is a pole (isolated singularity) instead of a non-singular point (Needham, 1998); The neighborhood around the pole can be represented by the series.

## References

Kramer, P. L.S. Examples. Retrieved August 22, 2020 from: http://eaton.math.rpi.edu/faculty/Kramer/CA13/canotes111113.pdf

Needham, T. (1998). Visual Complex Analysis. Clarendon Press.

Orloff, J. (2020). Topic 7 Notes. Retrieved August 22, 2020 from: https://math.mit.edu/~jorloff/18.04/notes/topic7.pdf

Stephenson, G. & Radmore, P. (1990). Advanced Methods for Engineering and Science. Cambridge University Press.

Complex Analysis III. Laurent Series and Singularities. Retrieved August 22, 2020 from: http://howellkb.uah.edu/MathPhysicsText/Complex_Variables/Laurent.pdf