An isolated singularity is essential when its Laurent series has infinitely many nonzero negative-power terms.
Near an isolated essential singularity, a holomorphic function has dense image in the complex plane. If it avoided a disk about , then would be bounded and extend across the singularity by the Riemann removable singularity theorem. If its extended value is nonzero, extends holomorphically; if zero, its finite-order zero makes a pole. Either contradicts an essential singularity.
Articles by others on the same topic
An **essential singularity** is a type of singular point in complex analysis that has specific properties. In a complex function \( f(z) \), a point \( z_0 \) is considered an essential singularity if the function behaves in a particularly wild manner as \( z \) approaches \( z_0 \). To understand this concept better, it's helpful to refer to the classification of singularities for complex functions.