Laurent series
Objective. Classify isolated singularities.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A Laurent series generalises Taylor series by allowing negative-power terms, writing f(z)=Σ_{n=-∞}^{∞}aₙ(z−z₀)ⁿ, valid on an annulus where f is holomorphic even if f is singular at z₀ itself. The coefficients are obtained via aₙ=(1/2πi)∮f(z)/(z−z₀)^{n+1}dz around a contour in the annulus. Isolated singularities are classified by the negative-power part: a removable singularity has no negative terms, a pole of order m has finitely many negative terms ending at a₋ₘ≠0, and an essential singularity has infinitely many negative-power terms. For example, f(z)=e^{1/z}=Σzⁿ/(-n)! for n≤0 has infinitely many negative terms about z=0, so z=0 is essential.
Worked example
Classify the singularity of the Laurent series f(z)=3/z² + 2/z + 1 + z at z=0.
- 1List the powers present: z⁻², z⁻¹, z⁰, z¹
- 2Negative powers stop at z⁻², so finitely many negative terms
- 3The lowest power is −2, giving a pole of finite order
- 4The singularity is a pole of order 2
Answer: Pole of order 2
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Start practisingCommon mistake. Students treat every singularity found by setting a denominator to zero as a pole without checking order via the Laurent expansion; sin(z)/z has an apparent singularity at z=0, but expanding gives 1−z²/6+..., with no negative powers, so z=0 is actually removable, not a pole.
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