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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).
The technological singularity —or simply the singularity[1] —is a hypothetical future point in time at which technological growth becomes uncontrollable and irreversible, resulting in unforeseeable consequences for Human civilization. [2][3] According to the most popular version of the singularity hypothesis, I. J. Good 's intelligence ...
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.
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.
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 ...
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 ...
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.
The Singularity Is Near: When Humans Transcend Biology. The Singularity Is Near: When Humans Transcend Biology is a 2005 non-fiction book about artificial intelligence and the future of humanity by inventor and futurist Ray Kurzweil. A sequel book, The Singularity Is Nearer, was released on June 25, 2024.