The residue theorem expresses a contour integral as a sum of complex residues.
Locally, let 
be analytic on 
, with Laurent
series
 |
(1)
|
which can be integrated term by term over a counterclockwise circle 
with center 
and radius less than 
,
 |  |  |
(2)
|
 |  |  |
(3)
|
For every integer 
, the function 
has a single-valued antiderivative
on the deleted neighborhood. The first and
last terms therefore vanish, giving
 |
(4)
|
where 
is the complex residue. Writing the contour
as 
,
where 
,
gives
 |
(5)
|
so we have
 |
(6)
|
More generally, let 
be a counterclockwise simple closed contour and let 
be analytic
on and inside 
except for finitely many isolated singularities
forming a set 
. Then
 |
(7)
|
The singularities may be poles or essential singularities. The value of the contour integral therefore depends only on the complex residues at the enclosed singularities.
The diagram above shows an example of the residue theorem applied to the illustrated contour 
and the function
 |
(8)
|
Only the poles at 1 and 
are contained in the contour, which have residues of 0 and
2, respectively. The value of the contour integral
is therefore given by
 |
(9)
|
