classification of singularities
Where an analytic function misbehaves — a point where it is undefined or blows up — it does so in one of a few sharply distinct ways. Some breakdowns are cosmetic and can be patched; some are violent but orderly; some are genuinely wild. Classifying singularities means sorting these isolated bad points into their types, because the type tells you everything about how the function behaves nearby and exactly how to integrate around it.
For an isolated singularity at a point a, the verdict comes from the Laurent series there. If the principal part (the negative-power terms) is absent, the singularity is removable: the function is secretly fine and can be given a value at a to make it analytic, as with (sin z)/z at z = 0, which simply equals 1. If the principal part has finitely many terms, ending at (z - a)^{-m}, you have a pole of order m: the function blows up like 1/(z - a)^m, cleanly and predictably. If the principal part has infinitely many negative-power terms, the singularity is essential, and the behaviour near a is chaotic — by the Casorati-Weierstrass picture, the function comes arbitrarily close to every complex value in any neighbourhood, as e^{1/z} does at the origin.
This taxonomy drives the whole computation. A removable singularity contributes nothing and can be ignored. A pole has a finite residue computed by a simple limit formula, and poles are the bread and butter of the residue theorem. An essential singularity still has a residue but resists the easy formulas, requiring the full Laurent expansion. Identifying the type first — before reaching for any residue formula — is what keeps applied contour integration honest and correct.
Three flavours at z = 0: (sin z)/z has Laurent series 1 - z^2/6 + ... with no negative powers, so the singularity is removable. The function 1/z^3 is a pole of order three. And e^{1/z} = 1 + 1/z + 1/(2 z^2) + ... has infinitely many negative powers, so z = 0 is an essential singularity.
Removable, pole, essential — read off directly from how many negative powers the Laurent series carries.
These categories apply to isolated singularities only. A branch point (where the function is multivalued, like the origin for sqrt(z)) is a different beast entirely — it is not isolated in the Laurent-series sense, has no single residue, and is handled with branch cuts, not the pole machinery.