# Riemann’s theorem on isolated singularities

Let the complex function $f(z)$ be holomorphic in a deleted neighbourhood of the point  $z=z_{0}$  of the closed complex plane $\mathbb{C}\cup\{\infty\}$.  This point is

• a regular point (or a removable singularity) iff $f(z)$ is bounded in a neighbourhood of $z_{0}$,

• a pole iff  $\displaystyle\lim_{z\to z_{0}}|f(z)|\,=\,+\infty$,

• an essential singularity iff there is neither of the above cases.

Title Riemann’s theorem on isolated singularities RiemannsTheoremOnIsolatedSingularities 2013-03-22 19:00:51 2013-03-22 19:00:51 pahio (2872) pahio (2872) 5 pahio (2872) Theorem msc 30E99 Riemann’s theorem RiemannsRemovableSingularityTheorem