defining a path
as an infinitesimal counterclockwise circle around the
point ,
and defining 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

(3)

so

(4)

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

(5)

Now, let ,
so .
Then

(6)

(7)

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

(8)

(9)

(10)

(11)

giving (1).

If multiple loops are made around the point , then equation (11) becomes