Residue calculus and real integrals

For a simple positively oriented piecewise smooth closed contour, assume f is meromorphic on an open neighborhood of the enclosed closed region, has no pole on the contour, and has only finitely many poles inside. Remove small disjoint circles around those poles. Cauchy's theorem on the remaining region relates the outer integral to the small-circle integrals. In each local Laurent series, every power except the negative first power integrates to zero, leaving 2πi times the residue.

The contour need not pass near a pole to register its contribution; the integral records enclosure. A reversed contour changes the sign. More general closed curves require winding numbers and appropriate domain hypotheses rather than counting every pole once. A singularity lying exactly on the integration path is excluded from the ordinary theorem. A principal-value prescription or an indented contour is a different problem whose extra contribution and limiting interpretation must be stated explicitly.

At a simple pole a, multiply by z−a and take the limit. If f is a quotient p/q with holomorphic numerator and denominator, q(a)=0, and q′(a)≠0, the residue is p(a)/q′(a). This formula assumes the denominator zero is simple; using it at a double zero would divide by zero and miss the needed expansion. Cancellation can also remove a proposed pole, so check the numerator before assigning an order.

For a pole of order m, write f as a holomorphic numerator h divided by $(z-a)^m$. The residue is the coefficient of degree m−1 in h's Taylor series, hence its derivative of that order divided by the factorial. This method and direct Laurent expansion are equivalent. A residue can be zero at a genuine higher-order pole, so the contour contribution and the singularity's severity remain distinct quantities.

To integrate a rational function over the whole real line, a large upper semicircle is often useful when no pole lies on the real axis. The closed integral equals the real segment integral plus the arc integral. If the rational function decays like one over the square of the modulus uniformly on the arc, its length grows only linearly, so the arc contribution tends to zero. State the actual denominator bound rather than inferring uniform complex decay from behavior on the real axis.

Exponential factors require particular care: a factor that decays in one half-plane can grow in the other, and an oscillatory real-axis expression does not select its contour by itself. Branch powers require a chosen branch and often a cut contour. In every case, specify the desired real integral's convergence, the contour orientation, enclosed singularities, and auxiliary limits. Residue arithmetic is usually the shortest part of a correct contour argument; the contour choice supplies the bridge to the real quantity.