Laurent series and isolated singularities

If f is holomorphic on an annulus $r<|z-a|<R$, it has a unique Laurent series there, converging locally uniformly on that annulus. The nonnegative powers form a Taylor-like part and the negative powers form the principal part. Coefficients are given by integrals around any positively oriented centered circle lying within the annulus. Cauchy's theorem makes the coefficient integral independent of the chosen radius while the circle stays inside that same analytic annulus.

The inner and outer radii matter. Expanding a rational factor by a geometric series requires the chosen ratio to have modulus less than one. A ratio appropriate inside a pole radius is invalid outside it, where rewriting in inverse powers yields another series. Neither algebraic identity overrides the convergence requirement. At the boundary circles, ordinary series convergence must be examined separately; the Laurent theorem guarantees the open annulus, not automatic convergence on its edges.

An isolated singularity at a means f is holomorphic on some full punctured disk $0<|z-a|<R$. If every negative coefficient vanishes, the missing value can be filled by the constant coefficient and the singularity is removable. If only finitely many negative coefficients occur and the most negative is −m, the point is a pole of order m. Then multiplication by $(z-a)^m$ produces a holomorphic function nonzero at a. Infinitely many nonzero negative coefficients define an essential singularity.

The coefficient of $(z-a)^{-1}$ is the residue. It is crucial for contour integration, but it does not by itself classify the singularity. A second-order pole can have zero residue, and an essential singularity can also have zero residue. For example, the exponential of 1/z² has infinitely many negative even powers and no negative first power. A branch point such as the origin for a continued logarithm is not an isolated singularity of one single-valued holomorphic function on a full punctured disk.

A Laurent coefficient computed on an annulus that surrounds several excluded points records the behavior on that annulus. Its negative first-power coefficient determines the integral on its centered circles, but it need not equal the local residue at the center. The local residue uses sufficiently small circles enclosing only the isolated central singularity. Crossing another pole changes the available annulus and can change the centered Laurent coefficients, even though the rational function itself is unchanged.

This distinction is useful when preparing a residue calculation. First list singularities and their distances from the chosen center. Then divide the plane into maximal circular annuli avoiding them, expand each factor using a convergent geometric ratio, and check the first few terms against the original expression. Classification of the central singularity uses the innermost punctured disk. Integrals on larger circles can be understood later as sums of local residues rather than silently assigning an outer coefficient to one point.