We are now ready to make the general statement:
If a function f(z) has a term
{R\over
z−c} in its Laurent series around the
point c (i.e., it is not analytic
in a region around c, but it has
an “isolated singularity” at c),
then for any contour that encloses this and only this pole
|
\mathop{∮
}\nolimits f(z)dz = ±2πiR
| (A.8) |
Here R is called the residue of f at c, and the sign depends on the orientation of the contour around c.
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!)
|
\mathop{∮
}\nolimits f(z)dz = ±2πi{\mathop{∑
}}_{k}{R}_{k}
| (A.9) |
We can find the residue by expanding f(z) around c; it is often more useful (quicker) to look at the limit
|
{\mathop{lim}}_{z→c}(z − c)f(z) = R.
| (A.10) |
This works if there are no higher order singularities, i.e. no terms {b}_{−2}∕{(z − c)}^{2}, etc. in the Laurent series.