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) |
(2) |
(3) |
(4) |
(5) |
(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) |
See also Maclaurin Series, Residue (Complex Analysis), Taylor Series
References
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.
© 1996-9 Eric W. Weisstein