Laurent series
/ loh-RAHN /
A Taylor series describes an analytic function near a point where it is perfectly well-behaved, using only nonnegative powers (z - a)^0, (z - a)^1, .... But what do you do near a point where the function blows up, like 1/z near the origin? The Laurent series is the answer: it is the Taylor series generalised to allow negative powers too, giving a local description even around a singularity. It is the microscope through which you read off the nature of any isolated singular point.
Around a point a, on an annular region (a ring where the function is analytic), the Laurent series is a sum from n = minus infinity to plus infinity of c_n (z - a)^n. The part with positive powers behaves like an ordinary Taylor series; the part with negative powers — the principal part — captures the blow-up. The single most important coefficient is c_{-1}, the coefficient of 1/(z - a): it is the residue, the only term that survives when you integrate the series around a loop, since the loop integral of (z - a)^n is zero for every n except n = -1, where it is 2 pi i. Everything in residue calculus comes down to extracting this one number.
The Laurent series is the diagnostic tool that classifies singularities: no negative powers means a removable singularity, finitely many means a pole, infinitely many means an essential singularity. In applied work you rarely compute it from the defining integral; instead you build it by reusing known expansions — the geometric series for 1/(1 - z), the exponential series, partial fractions — and read the residue straight off. That residue is what you feed into the residue theorem to crack real integrals and inverse transforms.
Expand f(z) = e^z / z^2 near z = 0. Use the exponential series e^z = 1 + z + z^2/2 + z^3/6 + ... and divide by z^2: f(z) = 1/z^2 + 1/z + 1/2 + z/6 + .... The principal part has two negative-power terms, so z = 0 is a pole of order two, and the residue (the coefficient of 1/z) is 1.
Reusing a known series and dividing gives the Laurent expansion and the residue at a glance.
A function has different Laurent series in different annuli around the same point — for instance one valid for small |z| and a totally different one for large |z|. You must state which ring you mean; reading the residue from the wrong annulus gives the wrong answer.