#
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
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