Wikipedia / Essential singularity
Wikipedia / Essential singularity
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior.
The category essential singularity is a "left-over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner removable singularities and poless. In practice some include non-isolated singularities too; those do not have a residue.
Formal description
Consider an open subset of the complex plane be an element of a holomorphic function is called an essential singularity if the singularity is neither a pole nor a removable singularity.
has an essential singularity at .
Alternative descriptions
be a complex number but is analytic of the complex plane, and that every open neighbourhood has non-empty intersection with .
is a removable singularity and .
does not exist (in fact is a pole and a zero of .
is an essential singularity and .
Another way to characterize an essential singularity is that the Laurent series has infinitely many negative degree terms (i.e., the principal part is not differentiable for any integer is an essential singularity of .
On a Riemann sphere with a point at infinity, has an essential singularity at exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, which is at . Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at .
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity takes on every never takes on the value .)