Pole (complex analysis)

| Main Article
| Discussion
| Related Articles  [?]
| Bibliography  [?]
| External Links  [?]
| Citable Version  [?]

|
|
|
|
|
|
|
|

This editable Main Article is under development and subject to a disclaimer.

[edit intro]

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

lim z -> a f ( z ) ( z − a ) k = r {\displaystyle \lim _{z\rightarrow a}f(z)(z-a)^{k}=r\,}

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

f ( z ) = ∑ n = − k ∞ c n ( z − a ) n , {\displaystyle f(z)=\sum _{n=-k}^{\infty }c_{n}(z-a)^{n},\,}

where the leading coefficient  c − k = r {\displaystyle c_{-k}=r} . The residue of f is the coefficient  c − 1 {\displaystyle c_{-1}} .

An isolated singularity may be either removable, a pole, or an essential singularity.

## References[edit]

* Tom M. Apostol (1974). Mathematical Analysis, 2nd ed. Addison-Wesley, 458.