Pole (complex analysis)
From Citizendium, the Citizens' Compendium
In complex analysis, a pole is a type of singularity of a function of a complex variable. In the neighbourhood of a pole, the function behave like a negative power.
A function f has a pole of order k, where k is a positive integer, at a point a if the limit
for some non-zero value of r.
The pole is an isolated singularity if there is a neighbourhood of a in which f is holomorphic except at a. In this case the function has a Laurent series in a neighbourhood of a, so that f is expressible as a power series
where the leading coefficient 
. The residue of f is the coefficient 
.
An isolated singularity may be either removable, a pole, or an essential singularity.
References
- Tom M. Apostol (1974). Mathematical Analysis, 2nd ed. Addison-Wesley, 458.
