## Laurent Series

Let there be two circular contours and , with the radius of larger than that of . Let be interior to and , and be between and . Now create a cut line between and , and integrate around the path , so that the plus and minus contributions of cancel one another, as illustrated above. From the Cauchy Integral Formula,

 (1)

Now, since contributions from the cut line in opposite directions cancel out,

 (2)

For the first integral, . For the second, . Now use the Taylor Expansion (valid for )
 (3)

to obtain

 (4)

where the second term has been re-indexed. Re-indexing again,

 (5)

Now, use the Cauchy Integral Theorem, which requires that any Contour Integral of a function which encloses no Poles has value 0. But is never singular inside for , and is never singular inside for . Similarly, there are no Poles in the closed cut . We can therefore replace and in the above integrals by without altering their values, so
 (6)

The only requirement on is that it encloses , so we are free to choose any contour that does so. The Residues are therefore defined by

 (7)

Arfken, G. Laurent Expansion.'' §6.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 376-384, 1985.
Morse, P. M. and Feshbach, H. Derivatives of Analytic Functions, Taylor and Laurent Series.'' §4.3 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 374-398, 1953.