Simply Connected Domains

Complex Analysis · Axiom Academy

LESSON Simply Connected Domains The "no holes" condition that decides whether a contour integral vanishes — the hypothesis behind Cauchy's theorem. 1. No Holes: The Shrinking-Loop Test Drop a closed loop anywhere in a region and try to shrink it to a single point without ever leaving the region. If you can always do this, the region is simply connected . A hole is exactly what stops you: a loop encircling it gets caught and cannot collapse past it. 2. Which Regions Pass the Test? Run the shrinking-loop test on the regions you meet constantly. The rule of thumb: anything convex or star-shaped passes; removing even one point or carving out a ring creates a hole and fails. Watch the test loop collapse (green) or catch (red) on each. Convex, no hole — every loop collapses. Simply connected. Convex and unbounded, still hole-free. Simply connected. Cutting a ray to infinity blocks any loop from circling 0 — the hole is gone. Simply connected. One missing point is a hole. A loop around 0 jams. Not simply connected. The inner boundary is a hole. Not simply connected. A disk with its center removed — still one hole. Not simply connected. The whole plane minus a point is not simply connected, but the whole plane minus a ray is. That extra cut is why can be defined as a single-valued analytic function on — no loop in that region can wind around the origin. 3. Why It Matters: Cauchy's Theorem

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