Mathematics.

singularities

Poles and Singularities

Complex Analysis30 minDifficulty6 out of 10

You should know: laurent series

Overview

An isolated singularity of f is a point z0 where f fails to be analytic but f is analytic on some punctured neighborhood 0 < |z − z0| < r. The Laurent series of f about z0 classifies the singularity into exactly one of three types: removable (no negative-power terms; f extends analytically to z0), a pole of order m (the principal part terminates, with a_{-m} ≠ 0 the lowest nonzero negative coefficient), or essential (infinitely many nonzero negative-power terms). Poles are the most common and useful type, since near a pole of order m, f(z) behaves like c/(z−z0)^m and |f(z)| → ∞ as z → z0. Essential singularities are far wilder: by the Casorati–Weierstrass theorem, f takes values arbitrarily close to every complex number in any neighborhood of an essential singularity.

Intuition

Think of the three singularity types as a ladder of increasing bad behavior. A removable singularity is really 'no singularity at all' — the function is bounded nearby and you can just fill in the missing value (like sin(z)/z at z=0). A pole is a controlled blow-up: f(z) behaves essentially like 1/(z-z0)^m, shooting off to infinity smoothly and predictably as z approaches z0 from any direction. An essential singularity is chaos: there is no simple algebraic blow-up rate, and the function's values swirl through (almost) every complex number infinitely often in any neighborhood of the point, as captured by the Casorati–Weierstrass and (stronger) Picard theorems.

Formal Definition

Definition

Let f have Laurent expansion f(z) = Σ_{n=-∞}^{∞} a_n(z-z0)^n on a punctured neighborhood of z0. The singularity at z0 is:

Removable: an=0 for all n<0\text{Removable: } a_n = 0 \text{ for all } n < 0
Removable singularity
Pole of order m:a−m≠0, an=0 for n<−m\text{Pole of order } m: a_{-m}\neq 0,\ a_n = 0 \text{ for } n < -m
Pole of order m
Essential: an≠0 for infinitely many n<0\text{Essential: } a_n \neq 0 \text{ for infinitely many } n < 0
Essential singularity
f(z)=g(z)(z−z0)m,g analytic,g(z0)≠0f(z) = \frac{g(z)}{(z-z_0)^m}, \qquad g \text{ analytic}, g(z_0) \neq 0

Equivalent characterization of a pole of order m

Pole via a nonvanishing analytic factor

Worked Examples

  1. Expand sin(z) as a Taylor series and divide by z.

    sin⁡zz=z−z3/6+z5/120−⋯z=1−z26+z4120−⋯\frac{\sin z}{z} = \frac{z - z^3/6 + z^5/120 - \cdots}{z} = 1 - \frac{z^2}{6} + \frac{z^4}{120} - \cdots

  2. The resulting Laurent series has no negative-power terms at all — it is actually a Taylor series.

    an=0 for all n<0a_n = 0 \text{ for all } n < 0

Answer: z0 = 0 is a removable singularity: defining f(0) = 1 extends f to an entire function.

Practice Problems

Difficulty 5/10

Classify the singularity of f(z) = (1-cos z)/z² at z0=0.

Difficulty 5/10

Classify the singularity of f(z) = 1/(z³ - z²) = 1/(z²(z-1)) at z0=0.

Difficulty 7/10

Classify the singularity of f(z) = e^{1/z²} at z0=0.

Quiz

A singularity z0 is removable when the Laurent series about z0 has:
sin(z)/z at z=0 is:
Near an essential singularity, by the Casorati–Weierstrass theorem, f(z):

Summary

  • Isolated singularities are classified via the Laurent series' principal part: removable (none), pole of order m (finitely many, lowest term a_{-m}), essential (infinitely many).
  • Near a pole of order m, f(z) behaves like c/(z-z0)^m and |f(z)| → ∞; near an essential singularity, behavior is far wilder (Casorati–Weierstrass).
  • Classifying singularities is a prerequisite to computing residues and applying the residue theorem to contour integrals.

References

  1. WebsiteWikipedia — Singularity (mathematics)