Start with an open subset *U* of the complex plane containing the number *z*_{0}, and a holomorphic function *f* defined on *U* - {*z*_{0}}. The complex number *z*_{0} is called an *essential singularity* if there is *no* natural number *n* such that the limit

The Weierstrass-Casorati theorem states that

**if***f*has an essential singularity at*z*_{0}, and*V*is any neighborhood of*z*_{0}contained in*U*, then*f*(*V*) is dense in**C**. Or spelled out: if ε > 0 and*w*is any complex number, then there exists a complex number*z*in*U*with |*z*-*z*_{0}| < ε and |*f*(*z*) -*w*| < ε.