The g Over h Rule for Evaluating Residues
Letwhere the functionsandare analytic at the pointand then
In order to prove the theorem we first need to show that an if an analytic functionhas a singularity atthenproviding this limit exists.
To do this letIfthenhas a removable singularity atIf then f has a simple pole atand
(Note that we can expandin a Laurent series aboutobtaining
Now putthen(since ), then
The theorem is proved.
Example: Evaluate the residue ofat
atandatso therule gives