The word "pole" is used prominently in a number of very different branches of mathematics. Perhaps the most important and widespread usage is to denote a singularity of a complex
function. In inversive geometry, the inversion pole is related to inverse
points with respect to an inversion circle.
The term "pole" is also used to denote the degenerate points 
and 
in spherical coordinates, corresponding
to the north pole and south
pole respectively. "All-poles method" is an alternate term for the
maximum entropy method used in deconvolution.
In triangle geometry, an orthopole
is the point of concurrence of certain perpendiculars with
respect to a triangle of a given line,
and a Simson line pole is similarly defined based
on the Simson line of a point
with respect to a triangle. In projective
geometry, the perspector is sometimes known as
the perspective pole.
In complex analysis, an analytic function 
is said to have a pole of order 
at a point 
if, in the Laurent series,
for 
and 
.
Equivalently, 
has a pole of order 
at 
if 
is a positive integer
for which 
extends to a holomorphic
function at 
with a nonzero value there. An analytic
function 
has a pole at infinity if
 |
A nonconstant polynomial 
has a pole at infinity of
order 
, i.e., the polynomial
degree of 
.
The basic example of a pole is 
,
which has a single pole of order 
at 
. Plots of 
and 
are shown above in the complex plane.
For a rational function written with coprime numerator and denominator,
the finite poles are the roots of the denominator,
where a root of multiplicity 
corresponds to a pole of order 
.
A holomorphic function whose only singularities are poles is called a meromorphic function.
Renteln and Dundes (2005) give the following (bad) mathematical jokes about poles:
Q: What's the value of a contour integral around Western Europe? A: Zero, because all the Poles are in Eastern Europe.
Q: Why did the mathematician name his dog "Cauchy?" A: Because he left a residue at every pole.
