A **pole** (also called an *isolated singularity*) is a point where the limit of a complex function inflates dramatically with polynomial growth.

## Graph of a Pole

The following graph of the absolute value of the gamma function shows several poles:

## A More Precise Definition

More specifically, a point z_{0} is a pole of a complex-valued function *f* if the function value *f*(z) tends to infinity as z gets closer to z_{0}. If the limit *does *exist, then the point is not a pole (it is a removable singularity).

A simple example is the **complex reciprocal function** 1/z, which has a pole at z = 0. Any function of the form f(z) = z^{n} has a pole of order n at z = 0.

## Order

Z_{0} is a pole of order n if:

What we’re doing here is multiplying the function f(z) by (z – z_{0})^{n} and then taking the limit as z approaches z_{0}(z → z_{0}). **If the result is not equal to zero, then you have a pole.**

A slightly different way to think about this is with a *Laurent series*. A Laurent series is a representation of the function as a power series, including negative degree terms.

A punctured disk*, 0 < |z – z_{0} | < can be written in terms of a *Laurent series*. Let’s say the series is:

Then we can say that z_{0} is a pole of order p.

*A punctured disk is a disk of radius r, with a central point a. An open disk is made up of a set of complex numbers less than *r*, while a punctured disk is the same disc, but with a removed. You can think of it as a disk with a pinprick in the center (a pinprick that only removes the exact center, *a*).

## References

Image: https://math.wikia.org/wiki/p o l e CC-BY-SA

Fontaine, F. (2011). Complex Numbers. Retrieved December 20, 2019 from: https://engfac.cooper.edu/pages/fred/uploads/complex0.pdf