now I would like to extended this out into the complex plane and using the Cauchy integral to find the value, but not sure on how to do so exactly and what bounds to choose.
I figure I'll have something like:
For a pole of order 1 of a function f, find this limit :
and it'll give the residue.
More generally... :
A singularity is a pole of a function f if there exists a positive integer n, such that :
, where A is a finite value. (n will be the order)
Then there is a direct formula giving the residue :