Let P(z) and Q(z) be univariate polynomials in a complex variable z, and let the polynomial degrees of P and Q satisfy deg(Q)>=deg(P + 2). Then integral_γ (P(z))/(Q(z)) d z | = | 2π i sum_(a_i element A) Res_(z = a_i) (P(z))/(Q(z)) | = | -2 π i sum_(b_i element B) Res_(z = b_i) (P(z))/(Q(z)), where γ is a simple closed clockwise-oriented contour, A is the set of roots of Q inside of γ, and B is the set of roots of Q outside of γ.

complex residue | contour | contour integral | Jacobian | residue theorem | Riemann sphere | root