Laurent series and singularities

A Taylor series only has non-negative powers of (z − z₀), which forces it to represent a function that's already holomorphic at z₀. A Laurent series drops that restriction and allows negative powers too, which is exactly what lets it represent a function on an annulus around a point where the function isn't holomorphic — the negative-power part (the "principal part") is precisely what captures the bad behaviour there.

The number of terms in that principal part is the whole classification scheme. Zero negative-power terms means the singularity is removable: the function actually has a limit at z₀, and defining f(z₀) to equal that limit makes the function holomorphic there after all — sin(z)/z at z = 0 is the standard example, since its Laurent series is just 1 − z²/6 + z⁴/120 − ⋯, with no negative powers at all. Finitely many negative-power terms means a pole, with the order of the pole being the most negative power that appears. Infinitely many negative-power terms means an essential singularity, where no redefinition or algebraic trick removes the bad behaviour — near an essential singularity, the function takes on every complex value (with at most one exception) infinitely often, no matter how small a neighbourhood you look in.

One subtlety worth being explicit about: which Laurent series you get for a given f depends on which annulus around z₀ you expand it on, not just on f and z₀ alone. A function with two singularities can have a different valid Laurent expansion on the annulus between them than on the annulus beyond the outer one — so "the" Laurent series of a function is really shorthand for "the Laurent series on this particular annulus."