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Example 1

As an example, consider the contour integral

where C is some simple closed curve about 0.

Let us evaluate this integral using a standard convergence result about integration by series. We can substitute the Taylor series for e z {\displaystyle e^{z}} into the integrand. The integral then becomes

Let us bring the 1/ z factor into the series. The contour integral of the series then writes

Since the series converges uniformly on the support of the integration path, we are allowed to exchange integration and summation. The series of the path integrals then collapses to a much simpler form because of the previous computation. So now the integral around C of every other term not in the form cz is zero, and the integral is reduced to

The value 1/4! is the residue of e / z at z = 0, and is denoted