residue

The residue is the single most important number attached to an isolated singularity — a distilled summary of all the function's misbehavior at that point, packed into one complex number. Intuitively it is the 'leftover' that survives when you integrate the function in a small loop around the singularity: every other piece of the function integrates away to nothing, and what remains, divided by 2pi i, is the residue. It is the residue, the residuum, the thing left behind.

Precisely, the residue of f at an isolated singularity z0 is the coefficient a_{-1} of the (z - z0)^{-1} term in the Laurent expansion of f around z0. Equivalently, it equals (1/(2pi i)) times the contour integral of f around any small loop enclosing z0 and no other singularity. For a simple pole, the residue is the limit as z -> z0 of (z - z0) f(z); for a pole of order m there is a derivative formula involving the (m-1)-st derivative of (z - z0)^m f(z).

Residues matter because of the residue theorem: a closed contour integral equals 2pi i times the sum of the residues enclosed. This converts the global problem of integrating around a loop into the purely local problem of computing a few coefficients. The technique evaluates real integrals, sums infinite series, and counts zeros and poles. A caveat: only the coefficient of the (-1) power is the residue — higher negative powers like a_{-2} contribute to the singularity's strength but integrate to zero and do not enter the residue.