singularity

A singularity is a point where a function's holomorphic good behavior breaks down — a flaw, gap, or blow-up in an otherwise smooth complex landscape. Often it is a single bad point surrounded by a region where the function is perfectly holomorphic (an isolated singularity), like a puncture in an otherwise pristine sheet. The whole art of residue calculus is about cataloguing and exploiting exactly these defects.

Precisely, a point z0 is a singularity of f if f fails to be holomorphic at z0 (but typically is holomorphic at points arbitrarily close to it). When f is holomorphic on a punctured disc around z0 but not at z0 itself, z0 is an isolated singularity, and these come in exactly three flavors classified by the Laurent series there: removable (the function can be redefined to be holomorphic, no negative powers), pole (finitely many negative powers, the function blows up to infinity), and essential (infinitely many negative powers, wild behavior).

The classification is sharp and complete: every isolated singularity is exactly one of those three. A removable singularity is no real obstruction — Riemann's theorem says a singularity is removable precisely when f stays bounded near z0. Be careful to distinguish isolated singularities from non-isolated ones (like a branch point of a square root, or the dense cluster of poles a function might have), which lie outside this neat trichotomy.