Contour integration is the process of calculating the values of a contour integral around a given contour in the complex plane. As a result of a truly amazing property of holomorphic functions, such integrals can be computed easily simply by summing the values of the complex residues inside the contour. Let P(x) and Q(x) be polynomials of polynomial degree n and m with coefficients b_n, ..., b_0 and c_m, ..., c_0. Take the contour in the upper half-plane, replace x by z, and write z congruent R e^(i θ). Then integral_(-∞)^∞ (P(z) d z)/(Q(z)) = lim_(R->∞) integral_(-R)^R (P(z) d z)/(Q(z)).

Cauchy integral formula | Cauchy integral theorem | complex residue | contour | contour integral | inside-outside theorem | Jordan's lemma | sine integral