Meromorphic function

Intuitively, a meromorphic function is one that is almost holomorphic. It behaves ideally throughout most of its domain, except at isolated points where it "blows up" in a controlled way. These controlled singularities are called poles.
Unlike essential singularities (which behave chaotically), poles are predictable: they simply send the point to infinity.

Definition (in the complex plane)

Formally, a function $f$ is said to be meromorphic on an open set $D\subseteq \mathbb{C}$ if there exists a set $P\subset D$ of isolated points such that:

• $f$ is holomorphic on $D\setminus P$, and
• every point $z_0\in P$ is a pole of $f$.

Examples:

• Rational Functions: Any quotient of polynomials, such as

is meromorphic on all of $\mathbb{C}$. Its poles are the roots of the denominator.

• Trigonometric Functions: The function

is meromorphic in $\mathbb{C}$, with poles at the zeros of $\cos (z)$, i.e.,

Another perspective: The Riemann Sphere ($\overline{\mathbb{C}}$)

This is where the concept reveals its true elegance. We can consider the following simplified definition:

A function on a domain $D\subseteq \mathbb{C}$ is meromorphic if it is a holomorphic map

where $\overline{\mathbb{C}}$ is the Riemann sphere. I.e. it is a holomorphic function that is allowed to take the value $\mathrm{\infty }$ at isolated points (its poles).

The domain $D$ can be generalized to be any Riemann surface.

This way, *poles are no longer singularities: A pole at $z_0$ is just a regular point in the domain that maps to $\mathrm{\infty }$ in the codomain.

The Fundamental Theorem of Meromorphic Functions

In the special case where $D=\overline{\mathbb{C}}$ (the whole Riemann sphere), the only such maps are the rational functions:

Theorem: A function is meromorphic on the entire Riemann sphere if and only if it is a rational function

$$f(z)=\frac{P(z)}{Q(z)},P,Q\in \mathbb{C}[z].$$

In particular, Moebius transformations correspond to the degree-1 rational maps

Why are they important?

Meromorphic functions provide the natural setting for many central theorems in complex analysis:

• The residue theorem is essentially a tool for integrating meromorphic functions.
• The Argument Principle connects the number of zeros and poles of a meromorphic function inside a curve.

By understanding them as holomorphic maps between spheres, not only are definitions simplified, but a deeper, more geometric insight into the structure of complex analysis is also gained.