Formally, consider an open subset *U* of the complex plane **C**, an element *a* of *U* and a holomorphic function *f* defined on *U* - {*a*}. The point *a* is called an *essential singularity* for *f* if it is neither a pole nor a removable singularity.

For example, the function *f*(*z*) = exp(1/*z*) has an essential singularity at *a* = 0.

The point *a* is an essential singularity if and only if the limit lim_{z→a} *f*(*z*) does not exist as a complex number nor equals infinity. This is the case if and only if the Laurent series of *f* at the point *a* has infinitely many negative degree terms.

The behavior of holomorphic functions near essential singularities is described by the Weierstrass-Casorati theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of *a*, the function *f* takes on *every* complex value, except possibly one, infinitely often.