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Plot of the function exp(1/z), centered on the essential singularity at z = 0.The hue represents the complex argument, the luminance represents the absolute value.This plot shows how approaching the essential singularity from different directions yields different behaviors (as opposed to a pole, which, approached from any direction, would be uniformly white).
Singularity (mathematics) In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. [1][2][3] For example, the reciprocal function has a singularity at , where the ...
A short proof of the theorem is as follows: Take as given that function f is meromorphic on some punctured neighborhood V \ {z 0}, and that z 0 is an essential singularity. . Assume by way of contradiction that some value b exists that the function can never get close to; that is: assume that there is some complex value b and some ε > 0 such that ‖ f(z) − b ‖ ≥ ε for all z in V at ...
Great Picard's Theorem (meromorphic version): If M is a Riemann surface, w a point on M, P1 (C) = C ∪ {∞} denotes the Riemann sphere and f : M \ {w} → P1 (C) is a holomorphic function with essential singularity at w, then on any open subset of M containing w, the function f (z) attains all but at most two points of P1 (C) infinitely often.
t. e. 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). Technically, a point z0 is a pole of a function f if it is a zero of the function 1/f and 1/f is ...
Removable singularity. 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. For instance, the (unnormalized) sinc function, as defined by.
Singularity theory. In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself ...
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. [3] An example of a transcendental branch point is the origin for the multi-valued function