Laurent Series

Key idea

Series with negative powers around singularities.

In depth

In an annulus around a singularity, f(z) = Σ aₙ(z − a)ⁿ over all integers n, including negative powers. The negative part captures the singular behaviour, and a₋₁ is the residue.

Worked example

1/(z(1 − z)) = 1/z + 1 + z + ... for 0 < |z| < 1.

Real-world uses

• Analysing signals and circuits with singular behaviour, such as poles in filter design

Common mistake

Using a Taylor expansion where the function is singular.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.