Let $U\subset\mathbb{C}$ be a domain, $a\in U$ and let $f:U \setminus \{a\} \to \mathbb{C}$ be holomorphic. If the Laurent series expansion of $f(z)$ around $a$ contains infinitely many terms with negative powers of $z-a$ then $a$ is said to be an essential singularity of $f$ Any singularity of $f$ is a removable singularity, a pole or an essential singularity.

If $a$ is an essential singularity of $f$ then the image of any punctured neighborhood of $a$ under $f$ is dense in $\mathbb{C}$ (the Casorati-Weierstrass theorem). In fact, an even stronger statement is true: according to Picard's theorem, the image of any punctured neighborhood of $a$ is $\mathbb{C}$ with the possible exception of a single point.