Meromorphic Functions

Meromorphic functions are the natural generalization of rational functions: they are holomorphic except for isolated poles. The theory of meromorphic functions connects to algebraic geometry through the study of divisors on Riemann surfaces.

Definition and Examples

Definition9.3Meromorphic function

A function $f$ on a domain $D$ is meromorphic if it is holomorphic on $D$ except for a set of isolated points, each of which is a pole. Equivalently, $f$ can be locally expressed as a ratio $g/h$ of holomorphic functions with $h \not\equiv 0$ .

ExampleExamples of meromorphic functions

Rational functions $P(z)/Q(z)$ are meromorphic on $\mathbb{C}$ (and on $\hat{\mathbb{C}}$ ).
$\tan z = \sin z / \cos z$ is meromorphic on $\mathbb{C}$ with poles at $z = (n + 1/2)\pi$ .
$\Gamma(z)$ is meromorphic on $\mathbb{C}$ with simple poles at $z = 0, -1, -2, \ldots$
The Weierstrass $\wp$ -function is meromorphic on $\mathbb{C}$ with double poles at lattice points.

Mittag-Leffler Theorem

Theorem9.3Mittag-Leffler theorem

Given a sequence of distinct points $\{a_n\} \subset \mathbb{C}$ with $|a_n| \to \infty$ and principal parts (Laurent tails) $p_n(z) = \sum_{k=1}^{m_n} c_{n,k}/(z-a_n)^k$ , there exists a meromorphic function $f$ on $\mathbb{C}$ whose only singularities are poles at $\{a_n\}$ with the prescribed principal parts.

RemarkNon-uniqueness

The function $f$ is not unique: if $f_1, f_2$ are two solutions, then $f_1 - f_2$ is entire. The Mittag-Leffler theorem is the additive analogue of the Weierstrass factorization theorem (which is multiplicative).

ExamplePartial fraction of $\cot(\pi z)$

The function $\pi\cot(\pi z)$ has simple poles at all integers with residue $1$ . By the Mittag-Leffler theorem:

$\pi\cot(\pi z) = \frac{1}{z} + \sum_{n=1}^{\infty}\left(\frac{1}{z-n} + \frac{1}{z+n}\right) = \frac{1}{z} + 2z\sum_{n=1}^{\infty}\frac{1}{z^2-n^2}.$

Setting $z = 0$ in the derivative yields the evaluation $\sum_{n=1}^\infty 1/n^2 = \pi^2/6$ .

Meromorphic Functions on the Riemann Sphere

Definition9.4Meromorphic on $\hat{\mathbb{C}}$

A meromorphic function on the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$ is one that is meromorphic on $\mathbb{C}$ and also meromorphic at $\infty$ (when expressed in the local coordinate $w = 1/z$ ). The meromorphic functions on $\hat{\mathbb{C}}$ are precisely the rational functions.

Theorem9.4Characterization of rational functions

$f$ is meromorphic on $\hat{\mathbb{C}}$ if and only if $f = P(z)/Q(z)$ for polynomials $P, Q$ . This is because a meromorphic function on a compact Riemann surface has finitely many poles, and the principal parts and polynomial growth at $\infty$ force $f$ to be rational.

Elliptic Functions

Definition9.5Elliptic function

An elliptic function is a meromorphic function on $\mathbb{C}$ that is doubly periodic: $f(z + \omega_1) = f(z + \omega_2) = f(z)$ for two $\mathbb{R}$ -linearly independent periods $\omega_1, \omega_2$ .

RemarkProperties of elliptic functions

Every non-constant elliptic function:

Must have poles (by Liouville's theorem, a holomorphic doubly periodic function is constant).
Has at least two poles (counted with multiplicity) in each fundamental parallelogram.
Satisfies: (sum of residues in a period parallelogram) = $0$ .
Takes every value in $\hat{\mathbb{C}}$ the same number of times (the order of the function).

ExampleThe Weierstrass $\wp$-function

For a lattice $\Lambda = \mathbb{Z}\omega_1 + \mathbb{Z}\omega_2$ , the Weierstrass $\wp$ -function is

$\wp(z) = \frac{1}{z^2} + \sum_{\omega \in \Lambda \setminus \{0\}}\left(\frac{1}{(z-\omega)^2} - \frac{1}{\omega^2}\right).$

It is elliptic of order $2$ and satisfies the differential equation $(\wp')^2 = 4\wp^3 - g_2\wp - g_3$ where $g_2, g_3$ are the Eisenstein invariants of $\Lambda$ .