Meromorphicity

Meromorphicity refers to the property of a complex-valued function being meromorphic on a given domain. A function f defined on an open subset U of the complex plane is meromorphic if it is holomorphic on U except at isolated points where f has poles. Equivalently, f can be written locally as a quotient f = g/h of holomorphic functions g and h on U, with h not identically zero. Another standard characterization is that around every point in U, f admits a Laurent expansion with only finitely many negative-power terms, so the singularities are poles rather than essential or removable.

At its poles, the function exhibits a well-controlled blow-up: if z0 is a pole of f, then (z − z0)^m f(z) extends holomorphically to z0 for some positive integer m, called the order of the pole. Points where f is holomorphic are regular, while removable singularities occur when the principal part of the Laurent series vanishes. If a singularity is not a pole, it is typically called essential and the function does not behave like a pole there.

Examples include f(z) = 1/z and f(z) = 1/(z^2 + 1), which are meromorphic on the complex plane with poles at their zeros; in contrast, sin(1/z) has an essential singularity at z = 0 and is not meromorphic there. Entire functions are meromorphic on the extended plane, and rational functions are precisely the meromorphic functions on the Riemann sphere.

Meromorphic functions form a field under addition and multiplication on connected domains, and they play a central role in complex analysis, residue theory, and the study of Riemann surfaces.