Computing a Principal Value

Complex Analysis · Axiom Academy

EXAMPLE Computing a Principal Value An indented contour handles a simple pole sitting right on the real axis. Evaluate the principal-value integral , then read off the real integrals and hiding inside it. The indented contour: the real axis from −R to R, dented up by a small semicircle C to skirt the pole at z = 0, closed by a large semicircle C R in the upper half-plane. Nice work. You used an indented contour to tame a pole sitting on the path of integration, then split one complex result into two real integrals. Indent, don't enclose: denting the contour around z=0 keeps the pole outside , so by Cauchy's theorem — no residue term on the right. A small arc gives half a residue: shrinking onto a simple pole, an arc of opening angle contributes — here for the clockwise upper indent. One integral, two answers: , so splits into (odd) and . The Dirichlet integral falls out: is even, so — a famous result, free of charge. Poles on the contour aren't obstacles — with a tiny detour they become exactly the terms that pin down real integrals classical methods can't touch.

This is the written version of the interactive lesson above. See the full Complex Analysis course.