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Borrowing from complex analysis, this is sometimes called an essential singularity. The possible cases at a given value for the argument are as follows. A point of continuity is a value of for which () = = (+), as one expects for a smooth function. All the values must be finite.
In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential 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 z 0 is an isolated singularity of a function f if there exists an open disk D centered at z 0 such that f is holomorphic on D \ {z 0}, that is, on the set obtained from D by taking z 0 out.
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 ...
A graph of a parabola with a removable singularity at x = 2. In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.
This function appears to have a singularity at z = 0, but if one factorizes the denominator and thus writes the function as = it is apparent that the singularity at z = 0 is a removable singularity and then the residue at z = 0 is therefore 0. The only other singularity is at z = 1.
Suppose that g is a global analytic function defined on a punctured disc around z 0.Then g has a transcendental branch point if z 0 is an essential singularity of g such that analytic continuation of a function element once around some simple closed curve surrounding the point z 0 produces a different function element.
The Vivanti–Pringsheim theorem is a mathematical statement in complex analysis, that determines a specific singularity for a function described by certain type of power series. The theorem was originally formulated by Giulio Vivanti in 1893 and proved in the following year by Alfred Pringsheim. More precisely the theorem states the following: