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Singularity functions are a class of discontinuous functions that contain singularities, i.e., they are discontinuous at their singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory. [1][2][3] The functions are notated with ...
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 ...
In mathematics, a real-valued function f on the interval [a, b] is said to be singular if it has the following properties: f is continuous on [a, b]. (**) there exists a set N of measure 0 such that for all x outside of N, the derivative f ′ (x) exists and is zero; that is, the derivative of f vanishes almost everywhere.
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).
In mathematics, the Gaussian or ordinary hypergeometric function 2F1 (a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE).
Cusp (singularity) A cusp at (0, 1/2) In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation.
Singular integral. In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator. whose kernel function K : Rn × Rn → R is singular along the diagonal x = y. Specifically, the singularity is such ...
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.