If 
is analytic
throughout the annular region between and on the concentric circles 
and 
centered at 
and of radii 
and 
respectively, then there exists a unique series expansion
in terms of positive and negative powers of 
,
 |
(1)
|
where
 |  |  |
(2)
|
 |  |  |
(3)
|
(Korn and Korn 1968, pp. 197-198).
Let there be two circular contours 
and 
, with the radius of 
larger than that of 
. Let 
be at the center of 
and 
, and 
be between 
and 
. Now create a cut line 
between 
and 
, and integrate around the path 
, so that the plus and minus contributions
of 
cancel one another, as illustrated above. From the Cauchy
integral formula,
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
Now, since contributions from the cut line in opposite directions cancel out,
 |  |  |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
For the first integral, 
.
For the second, 
.
Now use the Taylor series (valid for 
)
 |
(10)
|
to obtain
 |  | ![1/(2pii)[int_(C_1)(f(z^'))/(z^'-z_0)sum_(n=0)^(infty)((z-z_0)/(z^'-z_0))^ndz^'+int_(C_2)(f(z^'))/(z-z_0)sum_(n=0)^(infty)((z^'-z_0)/(z-z_0))^ndz^']](/images/equations/LaurentSeries/Inline52.svg) |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
where the second term has been re-indexed. Re-indexing again,
 |
(14)
|
Since the integrands, including the function 
, are analytic in the annular region defined by 
and 
, the integrals are independent of the path of integration
in that region. If we replace paths of integration 
and 
by a circle 
of radius 
with 
, then
 |  |  |
(15)
|
 |  |  |
(16)
|
 |  |  |
(17)
|
Generally, the path of integration can be any path 
that lies in the annular region and encircles 
once in the positive (counterclockwise) direction.
The complex residues 
are therefore defined by
 |
(18)
|
Note that the annular region itself can be expanded by increasing 
and decreasing 
until singularities of 
that lie just outside 
or just inside 
are reached. If 
has no singularities inside 
, then all the 
terms in (◇) equal zero and the Laurent series of
(◇) reduces to a Taylor series with coefficients
.
