A meromorphic function's Laurent series expansion about an isolated singularity yields a residue, defined as the coefficient of the (z - z₀)⁻¹ term, which equals (1/2πi) times the contour integral of the function over a small circle around that singularity. This belongs to complex analysis, specifically the theory of singularities and Laurent series, and residues are computed via specialized formulas depending on pole order (simple poles use limits or the p(z)/q'(z) rule; higher-order poles of order n use the (n-1)th derivative formula), with an extension defining the residue at infinity via a 1/z substitution.
A meromorphic function's Laurent series expansion about an isolated singularity yields a residue, defined as the coefficient of the (z - z₀)⁻¹ term, which equals (1/2πi) times the contour integral of…