Indented Contours

Complex Analysis · Axiom Academy

When a pole sits right on the real axis, you can't integrate through it — so you detour around it with a tiny semicircle, and that detour pays out exactly half a residue. 1. A Pole on the Path — So Detour Around It Suppose f(z) has a simple pole at a point x_0 on the real axis . The usual semicircular contour would have to run straight over the singularity — impossible. Instead we indent : we carve out a tiny semicircle of radius that hops over the pole, leaving the rest of the path untouched. The closed path is built in four pieces, traversed counter-clockwise: over the small semicircle around the pole, back over the large semicircle C_R . Then we let and . The straight pieces become the real integral we want; the arcs are what we have to evaluate. 2. A Semicircle Is Half a Residue A full small circle around a simple pole gives . The whole trick of the indent is that a semicircle sweeps only half the angle — so it gives exactly half of that. The animation runs the full loop, then the half loop, so you can watch become . A semicircle around a simple pole on the axis Why the sign and the factor. Near a simple pole, with g analytic. On write , so and the analytic part contributes nothing as : An upper indent is traced with (a span of ), giving . A lower indent runs over a span of , giving . Half the arc, half the residue — sign set by direction. 3. The Principal Value, and a Famous Payoff

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