Given a Contour Integral of the form
|
(1) |
define a path as an infinitesimal Circle around the point (the dot in the above illustration).
Define the path as an arbitrary loop with a cut line (on which the forward and reverse contributions cancel
each other out) so as to go around .
The total path is then
|
(2) |
so
|
(3) |
From the Cauchy Integral Theorem, the Contour Integral along any path not enclosing a Pole is 0.
Therefore, the first term in the above equation is 0 since does not enclose the Pole, and we are left
with
|
(4) |
Now, let
, so
. Then
But we are free to allow the radius to shrink to 0, so
and
|
(7) |
If multiple loops are made around the Pole, then equation (7) becomes
|
(8) |
where is the Winding Number.
A similar formula holds for the derivatives of ,
Iterating again,
|
(10) |
Continuing the process and adding the Winding Number ,
|
(11) |
See also Morera's Theorem
References
Arfken, G. ``Cauchy's Integral Formula.'' §6.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL:
Academic Press, pp. 371-376, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 367-372, 1953.
© 1996-9 Eric W. Weisstein
1999-05-26