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. 371376, 1985.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGrawHill, pp. 367372, 1953.
© 19969 Eric W. Weisstein
19990526