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) |

(4) |

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

(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) |

**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

1999-05-26