The Residue Theorem

Integrating a complex function around a closed loop sounds like it should depend on every twist of the path, but for an analytic function it depends on almost nothing: only on which poles the loop happens to encircle. That is the residue theorem, $\oint_C f(z)\,dz = 2\pi i \sum \mathrm{Res}(f, z_k)$ summed over the poles inside $C$, and it is the workhorse that turns hopeless real integrals into a quick tally of residues. The scene colours the plane by the phase of $f$, the hue cycling through every value around each pole so the singularities glow as little pinwheels, and marks each pole with its residue. Drag the contour around and resize it: the integral printed below is computed numerically from the loop, yet it always lands exactly on $2\pi i$ times the residues you have lassoed, and on nothing else. Slide the radius outward from zero and the integral sits flat while the loop is between poles, then jumps in a single step by $2\pi i\,\mathrm{Res}$ the instant a pole crosses the boundary, which the bottom panel records as a staircase. Enclose no poles and the loop integrates to zero; enclose every pole of a proper rational function and the residues cancel back to zero. Cycle the functions to watch the staircase rearrange itself around poles on the real axis, on the imaginary axis, and at the origin.