Cauchy's Theorem

Complex Analysis · Axiom Academy

The integral of an analytic function around any closed loop is zero — and the one situation where that fails. 1. Go All the Way Around — and Land at Zero Take a function f that is analytic everywhere inside a region with no holes, and any closed contour sitting in that region. As you travel once around , the contour integral accumulates contributions — but by the time you return to the start, they have perfectly cancelled . f analytic on a simply connected D , Write f = u + iv and expand with Green's theorem. The two area integrals that appear have integrands -v_x - u_y and u_x - v_y . The Cauchy–Riemann equations u_x = v_y , u_y = -v_x make both of them vanish — so the whole integral is zero. (Goursat later showed you don't even need f' to be continuous; hence the name Cauchy–Goursat theorem .) 2. Put a Hole Inside, and It No Longer Cancels Every word of the hypothesis earns its keep. Drop the function f(z) = 1/z , which has a singularity at the origin , and run the same trip around the unit circle. Now the accumulation does not return to zero — it winds up at . Cauchy's theorem isn't violated here — it simply doesn't apply . The domain on which 1/z is analytic is , and that domain is not simply connected: the loop encircles the missing point. Each of the three conditions is doing real work: Differentiable at every interior point — no poles, no branch cuts inside the loop. No holes: any loop in D can be shrunk to a point without leaving D .

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