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
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.