Formally, suppose *U* is an open subset of the complex plane **C**, *a* is an element of *U* and *f* : *U* − {*a*} → **C** is a holomorphic function. If there exists a holomorphic function *g* : *U* → **C** and a natural number *n* such that *f*(*z*) = *g*(*z*) / (*z* - *a*)^{n} for all *z* in *U* − {*a*}, then *a* is called a **pole of f**. If

The number *a* is a pole of order *n* of *f* if and only if the Laurent series expansion of *f* around *a* has only finitely many negative degree terms, starting with (*z* - *a*)^{−n}.

A pole of order 0 is a removable singularity. In this case the limit lim_{z→a} *f*(*z*) exists as a complex number. If the order is bigger than 0, then lim_{z→a} *f*(*z*) = ∞.

A non-removable singularity that is not a pole or a branch point is called an essential singularity.

A holomorphic function whose only singularities are poles is called meromorphic.