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

Proof:

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.