the Laurent coefficients

Once you accept that a function has a unique Laurent series on a given annulus, the natural question is: how do I actually find the numbers a_n? Those numbers, one for each integer n (positive, negative, and zero), are the Laurent coefficients. They package the function's local behaviour into a list, and there are two honest ways to get them: a clean integral formula that always works, and clever algebra that is usually faster.

The integral formula is the workhorse. For a function f holomorphic on the annulus r < |z - z_0| < R, the n-th coefficient is a_n = (1 / (2 pi i)) times the integral over gamma of f(z) / (z - z_0)^(n+1) dz, where gamma is any circle |z - z_0| = rho lying inside the annulus, traversed once counterclockwise. This is the Cauchy integral formula stretched to cover negative n as well; it works for every integer n at once and proves the expansion is unique. In practice, though, you rarely compute these integrals directly — instead you recognize known series. Multiply, divide, or substitute into the geometric series 1/(1 - w) = 1 + w + w^2 + ... or the exponential/sine/cosine series, and match powers of (z - z_0).

The most prized coefficient is a_{-1}: by setting n = -1 in the formula, a_{-1} = (1 / (2 pi i)) times the integral of f(z) dz over gamma, so the integral of f around the loop is exactly 2 pi i times a_{-1}. That single number is the residue, the bridge to the residue theorem and the evaluation of real integrals. The other coefficients matter too — a_{-m} being the lowest nonzero negative one tells you the pole has order m — but a_{-1} is the one that turns a series into a tool.