RESIDUE SHORTCUTS WITH EXAMPLES

We now mention some shortcuts that can sometimes be used to speed up the calculation of residues:

If has a pole of order at , then

Let’s justify this rule.  The Laurent series of f corresponding to a pole of order at is

So

Differentiating times yields

Letting gives the result.

As a particular case we have

If where and are analytic at , has a simple pole at , then (Notice that since is a simple zero of ).  Indeed, from above with m=1 have

.