An analytic function 
whose Laurent series
is given by
 |
(1)
|
can be integrated term by term using a closed contour 
encircling 
,
 |  |  |
(2)
|
 |  |  |
(3)
|
The Cauchy integral theorem requires that the first and last terms vanish, so we have
 |
(4)
|
where 
is the complex residue. Using the contour 
gives
 |
(5)
|
so we have
 |
(6)
|
If the contour 
encloses multiple poles, then the theorem gives the general result
 |
(7)
|
where 
is the set of poles contained inside the contour. This amazing theorem therefore
says that the value of a contour integral for
any contour in the complex plane depends
only on the properties of a few very special points inside the contour.
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 values of the contour integral
is therefore given by
 |
(9)
|
