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 $D \subseteq C$ if there exists a set $P \subset D$ of isolated points such that:

$f$ is holomorphic on $D ∖ P$ , and
every point $z_{0} \in P$ is a pole of $f$ .

Examples:

Rational Functions: Any quotient of polynomials, such as

f (z) = \frac{(z - i) (z + 5)}{z^{2} + 1},

is meromorphic on all of $C$ . Its poles are the roots of the denominator.

Trigonometric Functions: The function

f (z) = \tan (z) = \frac{\sin (z)}{\cos (z)}

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

z = \frac{π}{2} + k π, k \in Z .

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

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

A function is meromorphic if and only if it is a holomorphic map from the Riemann sphere to itself:

f : \bar{C} \to \bar{C} .

This way:

Poles are no longer singularities: A pole at $z_{0}$ is just a regular point in the domain that maps to $\infty$ in the codomain.
Infinity is just another point: The behavior as $z \to \infty$ is handled simply by evaluating $f (\infty)$ .
Unification: There are no special cases. The function’s behavior at any point (finite, pole, or infinity) is described within the same holomorphic framework, by choosing the suitable chart of $\bar{C}$ .

The Fundamental Theorem of Meromorphic Functions

This geometric view leads to one of the most beautiful and surprising results in complex analysis, connecting geometry with algebra:

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

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.