The coefficient of (z − z0)^{-1} in the Laurent expansion of a complex function about an isolated singularity z0; it captures the function's leading singular behavior relevant for contour integrals.

Principle

Local behavior near an isolated singularity is encoded by the Laurent series; the residue is the unique coefficient whose integral around a small loop equals 2πi times that coefficient.

Demonstration

For a simple pole at z0, Res_{z0} f = lim_{z→z0} (z−z0) f(z). For f(z)=1/(z−z0), the residue is 1 and ∮_C f(z) dz = 2πi.

Misapplication

Attempting to compute residues at branch points or non‑isolated singularities using the simple pole formula leads to incorrect results; residues require isolated singularities or appropriate branch cut handling.

Consequence

Residues allow exact evaluation of contour integrals and sums via the residue theorem, reduce complex integral computations to algebraic residue calculations, and classify singularity types by vanishing/nonvanishing residues.

Reversal

A zero residue at an isolated singularity does not necessarily mean the singularity is removable; it can be a higher‑order pole with symmetric coefficients canceling the (z−z0)^{-1} term.

Boundary

Defined only for isolated singularities (poles, removable singularities, essential singularities) in regions where a Laurent series exists; not defined for non‑isolated singularities or across branch cuts without specification.

Semantic Tension

Confused with coefficient extraction in Taylor series or with principal part: residue is specifically the (z−z0)^{-1} coefficient in the Laurent series, while other coefficients describe regular or higher‑order singular behavior.

Synthesis

A complex residue is the Laurent coefficient of order −1 at an isolated singularity that encapsulates the local singular contribution and determines the value of surrounding contour integrals.