Pole (complex analysis)

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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

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

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

${\displaystyle f(z)=\sum _{n=-k}^{\infty }c_{n}(z-a)^{n},\,}$

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

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.