Cauchy Integral theorem

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The Cauchy Integral theorem states that for a function which is analytic inside and on a simple closed curve in some region of the complex plane, for a complex number inside


For a constant, we may rewrite , and noting that and , and taking the contour of integration to be a circle of unit radius , we may write


By Cauchy's Theorem, we may deform the contour into any closed curve that contains the point and the result holds.

For the case when is not constant we may write


We must show that the second term on the left is identically zero.

In a vanishingly small neighborhood


Because the derivative of an analytic function is also analytic, the integral vanishes identically within a neighborhood of . By Cauchy's theorem, the contour of integration may be expanded to any closed curve within that contains the point thus showing that the integral is identically zero.