A
complex function is said to have a
pole at a point
a if there is an
isolated singularity at
a and the
Laurent series expansion around
a contains only
finitely many nonzero terms involving
negative powers of (
z-
a). (When the expansion contains no such terms, we have a
removable singularity rather than a pole.) If the Laurent series term with the most negative (i.e., minimum) exponent is (
z-
a)
-n we say that the function has a "pole of
order n" at this point. Poles of order 1 are sometimes referred to as "
simple" poles; for example, the function 1 /
z has a simple pole at the origin.
Poles play an important role in the theory of contour integration, since the residue of a isolated singularity is easiest to calculate if the singularity is a pole, making the Cauchy residue theorem particularly easy to apply.