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Can anyone please explain what Cauchy's residue theorem really means?

How can we integrate a function simply by adding its residues? I mean if we want to get an area over which the function is analytic we should remove the area where it is undefined, but according to the theorem the integration becomes 0 when there is no pole.

I'm not getting the idea, i searched for videos but each videos simply states theorem and shows its application, but where did it come from? I know this formula has been derived from Laurent (or even Taylor)-expansion, but why? I just want to know the physical explanation. Thanks in advance.

Leolime
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  • What is a "physical explanation"? In fact there is a physical explanation, or rather a physics-related analogy to conservative forces. Are you familiar with these? Essentially, certain well-behaved force fields can be written as the gradient of a scalar potential. And then by a generalization FTC, the integral between two points is the difference in this potential (regardless of path), so the integral over any closed loop is zero. Holomorphic functions have an antiderivative that plays the same role. – spaceisdarkgreen Jul 21 '17 at 13:26
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    Try Needham's Visual Complex Analysis for lots of geometric intuition and some physics. – lhf Jul 21 '17 at 13:53
  • See also https://math.stackexchange.com/questions/255612/intuitive-explanation-of-residue-theorem-in-complex-analysis. – lhf Jul 21 '17 at 13:54

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