
Isolated singularity
Has no other singularities close to it / From Wikipedia, the free encyclopedia
Dear Wikiwand AI, let's keep it short by simply answering these key questions:
Can you list the top facts and stats about Isolated singularity?
Summarize this article for a 10 years old
In complex analysis, a branch of mathematics, an isolated singularity is one that has no other singularities close to it. In other words, a complex number z0 is an isolated singularity of a function f if there exists an open disk D centered at z0 such that f is holomorphic on D \ {z0}, that is, on the set obtained from D by taking z0 out.
Mathematical analysis → Complex analysis |
Complex analysis |
---|
![]() |
Complex numbers |
Complex functions |
Basic theory |
Geometric function theory |
People |
Formally, and within the general scope of general topology, an isolated singularity of a holomorphic function a function is any isolated point of the boundary
of the domain
. In other words, if
is an open subset of
,
and
is a holomorphic function, then
is an isolated singularity of
.
Every singularity of a meromorphic function on an open subset is isolated, but isolation of singularities alone is not sufficient to guarantee a function is meromorphic. Many important tools of complex analysis such as Laurent series and the residue theorem require that all relevant singularities of the function be isolated.
There are three types of isolated singularities: removable singularities, poles and essential singularities.