Example
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 into the integrand. The integral then becomes
Let us bring the 1/z5 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: recall that
So now the integral around C of every other term not in the form cz−1 is zero, and the integral is reduced to
The value 1/4! is the residue of ez/z5 at z = 0, and is denoted
Read more about this topic: Residue (complex Analysis)
Famous quotes containing the word example:
“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)