Mathematics.

singularity theory

Meromorphic Functions

Complex Analysis45 minDifficulty7 out of 10

You should know: complex differentiation, laurent series

Overview

A meromorphic function on an open set U ⊂ ℂ is one that is holomorphic on U except at a discrete set of poles. At each pole, the function tends to infinity but only like 1/(z−a)^n for some finite n (the order of the pole). Meromorphic functions can be thought of as 'holomorphic functions that are allowed to blow up nicely.' The set of meromorphic functions on a domain forms a field under pointwise addition and multiplication. Key examples include rational functions, tan z, cot z, sec z, and the Gamma function.

Intuition

A meromorphic function is like a holomorphic function but with 'controlled explosions' at isolated points. Near a pole of order m, the function looks like c/(z−p)^m — blowing up at a predictable rate. Unlike essential singularities (where the behavior is wild, by Picard's theorem), poles are tame: they can be removed by multiplying by (z−p)^m. On the Riemann sphere, a meromorphic function becomes a genuine holomorphic map to ℂ ∪ {∞}, so poles are just 'points mapping to ∞.'

Formal Definition

Definition

A function f: U → ℂ is meromorphic on U if every point of U is either a regular point (f holomorphic there) or a pole of finite order:

f meromorphic on U  ⟺  f holomorphic on U∖P, P discrete, each p∈P a polef \text{ meromorphic on } U \iff f \text{ holomorphic on } U \setminus P,\ P \text{ discrete, each } p \in P \text{ a pole}
Definition
f(z)=∑n=−m∞an(z−p)n,a−m≠0, m≥1f(z) = \sum_{n=-m}^{\infty} a_n (z-p)^n,\quad a_{-m} \neq 0,\ m \geq 1
Laurent expansion at a pole of order m
ordp(f)=−m (negative integer at a pole of order m)\text{ord}_p(f) = -m \text{ (negative integer at a pole of order } m\text{)}
Order at a pole

Worked Examples

  1. The poles are the zeros of the denominator z²(z²+1) = z²(z−i)(z+i).

    z2(z2+1)=0⇒z=0 (order 2), z=±i (order 1)z^2(z^2+1) = 0 \Rightarrow z = 0\ (\text{order }2),\ z = \pm i\ (\text{order }1)

  2. At z=0: the denominator vanishes to order 2, so f has a pole of order 2.

    ord0(f)=−2\text{ord}_0(f) = -2

  3. At z=i and z=−i: each factor (z−i) or (z+i) vanishes to order 1, so f has simple poles there.

    ord±i(f)=−1\text{ord}_{\pm i}(f) = -1

Answer: f has a pole of order 2 at z=0 and simple poles at z=±i.

Practice Problems

Difficulty 6/10

Is tan z meromorphic on ℂ? If so, find its poles and their orders.

Difficulty 7/10

Compute the residue of cot(πz) at z = n (n ∈ ℤ).

Difficulty 7/10

A function f is meromorphic on ℂ and |f(z)| ≤ |z|² for all z with |z| large. What can you conclude?

Common Mistakes

Common Mistake

A function with an essential singularity is meromorphic.

Essential singularities are not poles; meromorphic functions only have poles (where the Laurent expansion has finitely many negative-power terms).

Common Mistake

The sum/product of meromorphic functions may not be meromorphic.

The meromorphic functions on a domain form a field: sums, differences, products, and quotients (when denominator is not identically 0) are all meromorphic.

Quiz

What distinguishes a pole from an essential singularity?
On the Riemann sphere ℂ ∪ {∞}, a meromorphic function becomes:

Summary

  • A meromorphic function is holomorphic except at a discrete set of poles.
  • At a pole of order m, the Laurent series starts with a_{-m}(z−p)^{−m}, a_{-m}≠0.
  • The residue at a simple pole p is lim_{z→p}(z−p)f(z).
  • Meromorphic functions on a domain form a field.
  • On the Riemann sphere, meromorphic functions are holomorphic maps to ℂ ∪ {∞}.

References

  1. WebsiteWikipedia — Meromorphic function