# Pole (complex analysis)

From Citizendium

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.