Cauchys Theory of Residues

Using these, we can write the closed curve using definite limits 6 dd) = f idd = 2vi 

This is one of the fundamental results of contour integration and will find widespread applications the point here being that the enclosure of a simple pole at the origin always yields 2tt/. 

Suppose we perform the same test for a higher order pole, say 1 /s, again with r = 1 

In fact, if we perform the same test with the power relationship /(s) = s , where n is integer 

if n is a positive integer, or zero, the above is obviously in accordance with Cauchy s First theorem, since then fis) = s is analytic for all finite values of s. However, if n becomes a negative integer, the s is clearly not analytic at the point s = 0. Nonetheless, the previous result indicates the closed integral vanishes even in this case, provided only that n — 1. Thus, only the simple pole at the origin produces a finite result, when the origin is enclosed by a closed contour. 

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