isolated singularity

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 is an isolated singularity of a function if there exists an open disk centered at such that f is holomorphic on , that is, on the set obtained from by removing . Formally, and within the general scope of general topology, an isolated singularity of a holomorphic function is any isolated point of the boundary \partial \Omega of the domain . In other words, if U is an open subset of , and is a holomorphic function, then a is an isolated singularity of . Every singularity of a meromorphic function on an open subset U\subset \mathbb{C} 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. Isolated singularities may be classified into three distinct types: removable singularities, poles and essential singularities.

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 is an isolated singularity of a function if there exists an open disk centered at such that f is holomorphic on , that is, on the set obtained from by removing . Formally, and within the general scope of general topology, an isolated singularity of a holomorphic function is any isolated point of the boundary \partial \Omega of the domain . In other words, if U is an open subset of , and is a holomorphic function, then a is an isolated singularity of . Every singularity of a meromorphic function on an open subset U\subset \mathbb{C} 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. Isolated singularities may be classified into three distinct types: removable singularities, poles and essential singularities. The function \tan\left(\frac{1}{z}\right) is meromorphic on , with simple poles at , for every . Since , every punctured disk centered at 0 has an infinite number of singularities within it, so no Laurent expansion is available for around , which is in fact a cluster point of its poles. The function has a singularity at that is not isolated, since there are additional singularities at the reciprocal of every integer, which are located arbitrarily close to (though the singularities at these reciprocals are themselves isolated). The function defined via the Maclaurin series converges inside the open unit disk centred at and has the unit circle as its natural boundary.

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