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 – z0 | < can also be written in terms of a Laurent series. Let’s say the series is:
Then we can say that z0 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.
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 – z0. 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.
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