Deformation of Contours

Complex Analysis · Axiom Academy

LESSON Deformation of Contours Why you can bend, shrink, and reshape an integration path — and the one obstacle that stops you. Think of the integration path as a rubber band stretched between two pins. Inside a region where f is analytic you may stretch and slide it freely. The Deformation Theorem says every shape you bend it into gives the exact same integral — the value depends on the endpoints and the region, not on the wiggles in between. Two homotopic paths in D (same endpoints) 2. Singularities Are Obstacles Deformation only works while the path stays in the analytic region. A singularity is a hole punched in that region — a place the rubber band cannot pass through. Try to shrink a loop that encircles a pole and it snags : it can tighten but never collapse, so its integral stays stuck at a nonzero value. Nothing blocks it — the loop shrinks to a point, so the integral is 0 . The pole blocks the collapse; the loop snags and the integral cannot reach 0 . f is undefined at the singularity, so any valid path must keep a wide berth. This single obstruction is the seed of residues and the entire residue calculus. 3. Concentric Circles & Simply Connected Regions

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