Calculus How To

Laurent Series

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

laurent series definition

A Laurent series is defined with respect to a point c and path of integration γ (yellow) which lies in the annulus (blue).


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.

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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.).

principal part of the laurent series

Principal part of the Laurent series.



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:
order of pole

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.

Formal Definition

If a function f(z) is analytic on the annulus:
laurent series 1

The function can be represented by (Orloff, 2020):
laurent series 2

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.

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

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Stephanie Glen. "Laurent Series" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/laurent-series/
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