Pole
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 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 the smallest
positive integer for which 
is
holomorphic at 
. A 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, the poles are simply given by 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.
Contact the MathWorld Team
residues
