We are now ready to make the general
statement:
If a function
has a term
in its Laurent series around the point
(i.e., it is not analytic in a region around
, but it has an “isolated singularity” at
), then for any contour that encloses
this and only this pole
(A.8) |
Here is called the residue of at , and the sign depends on the orientation of the contour around .
If multiple singularities are enclosed, we find that (all residues contribute with the same sign, since the contour must enclose them with the same orientation!)
(A.9) |
We can find the residue by expanding around ; it is often more useful (quicker) to look at the limit
(A.10) |
This works if there are no higher order singularities, i.e. no terms , etc. in the Laurent series.