Multiply Connected Domains

Complex Analysis · Axiom Academy

LESSON Multiply Connected Domains When a domain has a hole, a contour integral no longer has to be zero — and the winding number is what controls its value. A region is simply connected if every closed curve inside it can be shrunk continuously to a point without ever leaving the region. A multiply connected region has one or more holes — excluded points that a loop can get caught on, so it can never collapse. 2. The Winding Number Counts the Wraps To turn "how the loop sits around the hole" into a number, count how many full revolutions the contour makes around the point. That count is the winding number , and it is captured by a single integral. Always a whole number: +1 per counterclockwise loop . The spoke from the point sweeps through a full . . Same path, opposite direction, opposite sign. . The spoke wobbles but never completes a turn. . Wrap twice and the count doubles. 3. Deform the Contour — the Integral Won't Budge Here is the payoff. As long as you do not cross the hole, you can deform one contour into another and the integral is unchanged. For f(z) = 1/z around the hole at the origin, every loop with winding number 1 gives the same value — no matter its shape or size. Why it holds: two contours that wrap the hole the same way are homotopic in the region where f is analytic, and Cauchy's theorem makes homotopic contours agree — the bridge between them encloses no singularity, so it contributes nothing.

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