The g Over h Rule for Evaluating Residues
Theorem
Let
where the functions
and
are analytic at the point
and 
then
Proof:
In order to prove the theorem we first need to show that an if an analytic function
has a singularity at
then
providing this limit exists.
To do this let
If
then
has a removable singularity at
If 
then f has a simple pole at
and
(Note that we can expandin a Laurent series aboutobtaining
so that
then
)
Now putthen
(since 
), then
The theorem is proved.
Example: Evaluate the residue of
at
at
and
atso the
rule gives
