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So folks, I'm learning complex analysis right now and I've come across one thing that simply fails to enter my mind: the Cauchy Integral Theorem, or the Cauchy-Goursat Theorem. It says that, if a function is analytic in a certain (simply connected) domain, then the contour integral over a simple closed path must be zero. My question is:Why? Sure, I've seen the proof, but I don't get the intuition, I can't visualize it. Can someone help me out with the intuition?