## Cauchy Integral Formula

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
 (5)

But we are free to allow the radius to shrink to 0, so
 (6)

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 ,

 (9)

Iterating again,
 (10)

Continuing the process and adding the Winding Number ,
 (11)

Arfken, G. Cauchy's Integral Formula.'' §6.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 371-376, 1985.