the residue at infinity

The complex plane has a natural extra point — infinity — that you reach by going arbitrarily far out in any direction (the Riemann sphere makes this precise by capping the plane with a single point at the top). A function can have a singularity at infinity just as it has singularities at finite points, and it turns out infinity carries its own residue, defined to keep the whole bookkeeping consistent.

The definition: the residue of f at infinity is Res(f, infinity) = minus (1 / (2 pi i)) times the integral over a large circle |z| = R of f(z) dz, where the circle is traversed counterclockwise. The minus sign and the orientation are deliberate: from infinity's point of view, a counterclockwise loop in the plane actually goes clockwise around infinity, so the sign restores the natural orientation as seen from the sphere's north pole. A clean computational formula follows from the substitution w = 1/z, which maps infinity to 0: Res(f, infinity) = minus the residue at w = 0 of (1 / w^2) f(1/w). In particular, a function can be perfectly holomorphic at infinity (no pole there) yet still have a nonzero residue at infinity — for instance 1/z, which is holomorphic at infinity but whose residue at infinity is -1.

Why bother: the residue at infinity is the device that lets you trade an awkward calculation involving many finite poles for a single calculation at infinity, and it is the missing piece that makes the sum of all residues add up to zero. It is the natural way to evaluate the dogbone-type integrals, where the function is single-valued outside a finite cut and the residue at infinity captures everything 'outside.' The subtle point to hold onto: being regular at infinity is not the same as having residue zero there — that surprising 1/z example is the canonical warning.