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Class of discontinuous functions
Singularity functions are a class of discontinuous functions that contain singularities, i.e., they are discontinuous at their singular points. Singularity
Singularity_function
Type of function
Mathematical singularity Generalized function Distribution Minkowski's question-mark function (**) This condition depends on the references "Singular function",
Singular_function
Continuous function that is not absolutely continuous
the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase, the Cantor staircase function, and the Cantor–Lebesgue
Cantor_function
Point where a mathematical object behaves irregularly
to the derivative, not to the original function. A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate
Singularity_(mathematics)
Has no other singularities close to it
holomorphic function, then a {\displaystyle a} is an isolated singularity of f {\displaystyle f} . Every singularity of a meromorphic function on an open
Isolated_singularity
Hypothetical event
The technological singularity, often simply called the singularity, is a hypothetical event in which technological growth accelerates beyond human control
Technological_singularity
Location around which a function displays irregular behavior
essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category essential singularity is a "left-over"
Essential_singularity
Undefined point on a holomorphic function which can be made regular
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
Removable_singularity
Class of mathematical function
singularity. The function f ( z ) = sin 1 z {\displaystyle f(z)=\sin {\frac {1}{z}}} is not meromorphic either, as it has an essential singularity at
Meromorphic_function
2005 non-fiction book by Ray Kurzweil
embraces the term "the singularity", which was popularized by Vernor Vinge in his 1993 essay "The Coming Technological Singularity." Kurzweil describes
The_Singularity_Is_Near
Mathematical theory
mathematical singularity as a value at which a function is not defined. For that, see for example isolated singularity, essential singularity, removable
Singularity_theory
Point on a curve not given by a smooth embedding of a parameter
singular point at the origin. However, a node such as that of y 2 − x 3 − x 2 = 0 {\displaystyle y^{2}-x^{3}-x^{2}=0} at the origin is a singularity of
Singular_point_of_a_curve
Phenomenon within general relativity
curvature singularity at the Cauchy horizon known as the mass-inflation singularity, the Cauchy horizon singularity, the infalling singularity, or the "fat
Mass_inflation
Type of function in mathematics
the nearest singularity is at z = − 1 {\displaystyle z=-1} . Complex singularities can determine the radius of convergence even for functions that are smooth
Analytic_function
Growth function exhibiting a singularity at a finite time
singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function 1
Hyperbolic_growth
Attribute of a mathematical function
residue theorem. The residue of a meromorphic function f {\displaystyle f} at an isolated singularity a {\displaystyle a} , often denoted Res
Residue_(complex_analysis)
Mathematical function having a characteristic S-shaped curve or sigmoid curve
for sigmoid functions not evident or intuitive M1: Inverse of singularity functions M2: Sigmoid functions of embedded positive functions M3: Rising a
Sigmoid_function
Point on a curve where motion must move backwards
such a singularity is in the same differential class as the cusp of equation x 2 − y 5 = 0 , {\displaystyle x^{2}-y^{5}=0,} which is a singularity of type
Cusp_(singularity)
Solution of a confluent hypergeometric equation
two of the three regular singularities merge into an irregular singularity. The term confluent refers to the merging of singular points of families of differential
Confluent hypergeometric function
Confluent_hypergeometric_function
Domain of convergence of power series
At z = 0, there is in effect no singularity since the singularity is removable. The only non-removable singularities are therefore located at the other
Radius_of_convergence
Inputs for which a function's value is non-zero
In mathematics, the support of a real-valued function f is the subset of the function's domain consisting of those elements that are not mapped to zero
Support_(mathematics)
Concept in differential equation mathematics
coefficients are analytic functions, and singular points, at which some coefficient has a singularity. Then amongst singular points, an important distinction
Regular_singular_point
Functions in mathematics
entire function will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however
Harmonic_function
Concept in complex analysis
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
Zeros_and_poles
Hypothetical phenomenon
In general relativity, a naked singularity is a hypothetical gravitational singularity without an event horizon. When there exists at least one causal
Naked_singularity
Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution
Softmax_function
Branch of mathematics studying functions of a complex variable
"pole" (or isolated singularity) of a function is a point where the function's value becomes unbounded, or "blows up". If a function has such a pole, then
Complex_analysis
Function in quantum field theory showing probability amplitudes of moving particles
In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one
Propagator
Special mathematical function defined as sin(x)/x
cases, the value of the function at the removable singularity at zero is understood to be the limit value 1. The sinc function is then analytic everywhere
Sinc_function
Theorem in complex analysis
to the nearest non-removable singularity; if there are no singularities (i.e., if f {\displaystyle f} is an entire function), then the radius of convergence
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
Method for load calculation in construction
A well organized family of functions called singularity functions are often used as a shorthand for the Dirac function, its derivative, and its antiderivatives
Euler–Bernoulli_beam_theory
Concept of complex analysis
choice of which method to use depends on the function in question, and on the nature of the singularity. According to the residue theorem, we have: Res
Residue_theorem
Complex-differentiable (mathematical) function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood
Holomorphic_function
Power series with negative powers
x} except at the singularity x = 0 {\displaystyle x=0} . More generally, Laurent series can be used to express holomorphic functions defined on an annulus
Laurent_series
General relativity model near spacetime singularities
relativity has a page on the topic of: BKL singularity A Belinski–Khalatnikov–Lifshitz (BKL) singularity is a model of the dynamic evolution of the universe
BKL_singularity
Extension of the domain of an analytic function (mathematics)
definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where
Analytic_continuation
Concept in algebraic geometry
does not is given by the isolated singularity of x2 + y3z + z3 = 0 at the origin. Blowing it up gives the singularity x2 + y2z + yz3 = 0. It is not immediately
Resolution_of_singularities
Association of one output to each input
if one follows a closed loop around a singularity. This jump is called the monodromy. The definition of a function that is given in this article requires
Function_(mathematics)
Advertising executive, published author and speaker
hotel that decides to replace its doorman. One may consider their singular function to be opening and closing the door which is easily replaceable by
Rory Sutherland (advertising executive)
Rory_Sutherland_(advertising_executive)
Function that is holomorphic on the whole complex plane
non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental
Entire_function
Structural design tool
are required. Bending Euler–Bernoulli beam theory Bending moment Singularity function#Example beam calculation "Simply Supported Beam – Shear and Moment
Shear_and_moment_diagram
Matrix decomposition
there is a universal constant that characterizes the regularity or singularity of a problem, which is the system's "condition number" κ=σmax/σmin
Singular_value_decomposition
Function with unusual fractal properties
In mathematics, Minkowski's question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904
Minkowski's question-mark function
Minkowski's_question-mark_function
&x<a\\1,&x\geq a.\end{cases}}} Singularity function Lecture 12: Beam Deflections by Discontinuity Functions. Introduction to Aerospace Structures
Macaulay_brackets
Mathematical function
The singularity spectrum is a function used in multifractal analysis to describe the fractal dimension of a subset of points of a function belonging to
Singularity_spectrum
not dominate the polar divisor of a non constant function. Riemann further proved the Riemann singularity theorem, identifying the multiplicity of a point
Theta_divisor
problem fails to have a unique solution need not be singular functions. In some cases, the term singular solution is used to mean a solution at which there
Singular_solution
Mathematical theorem
a removable singularity of f . Both possibilities contradict the assumption that the point z0 is an essential singularity of the function f . Hence the
Casorati–Weierstrass_theorem
Invariant that plays a role in algebraic geometry and singularity theory
particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If f is a complex-valued holomorphic function germ
Milnor_number
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
Analytic function in mathematics
infinity on the Riemann sphere the zeta function has an essential singularity. For sums involving the zeta function at integer and half-integer values, see
Riemann_zeta_function
Theorem about the range of an analytic function
lacunary value of the function. Great Picard's Theorem: If an analytic function f {\textstyle f} has an essential singularity at a point w {\textstyle
Picard_theorem
One-dimensional complex manifold
to two, via the exponential map (which is entire and has an essential singularity at infinity, so not defined at infinity, and misses zero and infinity)
Riemann_surface
Family of solutions to related differential equations
The Bessel function of the first kind is an entire function if α is an integer, otherwise it is a multivalued function with singularity at zero. The
Bessel_function
Second-order partial differential equation
conjugate harmonic functions. This construction is only valid locally, or provided that the path does not loop around a singularity. For example, if r
Laplace's_equation
Description of the degeneracy of a function
and in particular singularity theory, an Ak singularity, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced
Ak_singularity
Analytic function in mathematics
has a singularity at a point z when za = 1, and also when za2 = 1. By the induction suggested by the above equations, f must have a singularity at each
Lacunary_function
Theorem in complex analysis
entire function f {\displaystyle f} is bounded in a neighborhood of ∞ {\displaystyle \infty } , then ∞ {\displaystyle \infty } is a removable singularity of
Liouville's theorem (complex analysis)
Liouville's_theorem_(complex_analysis)
Integral transform and linear operator
Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t).
Hilbert_transform
Provides integral formulas for all derivatives of a holomorphic function
statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary
Cauchy's_integral_formula
Functions in harmonic analysis mathematics
singular along the diagonal x = y {\displaystyle x=y} . Specifically, the singularity is such that | K ( x , y ) | {\displaystyle |K(x,y)|} is of size | x
Singular_integral
Topics referred to by the same term
production by Santa Clara Vanguard Drum and Bugle Corps a singular function in mathematics Cantor function Baguenaudier, a disentanglement puzzle This disambiguation
Devil's_staircase
mathematical statement in complex analysis, that determines a specific singularity for a function described by certain type of power series. The theorem was originally
Vivanti–Pringsheim_theorem
Solutions of Legendre's differential equation
degree, Legendre functions of the first kind reduce to Legendre polynomials, which are bounded on [-1, 1]. It can be shown that the singularity of the Legendre
Legendre_function
Function defined by a hypergeometric series
hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as specific
Hypergeometric_function
Expression which is not assigned an interpretation
undefined, is called a singularity. Some different types of singularities include: Removable singularities – in which the function can be extended holomorphically
Undefined_(mathematics)
Real function with secant line between points above the graph itself
(−∞,∞), because of the singularity at x=0. LogSumExp function, also called softmax function, is a convex function. The function −logdet(X) on the domain
Convex_function
Point where the derivative of a function is zero or undefined (in certain cases)
connected components by a function of the degrees of the polynomials that define the variety. Singular point of a curve Singularity theory Nullcline Milnor
Critical_point_(mathematics)
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Area of mathematics
dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena
Catastrophe_theory
Geometric representation of the complex numbers
arctangent function takes the values (−π/2, π/2) (in radians), and some care must be taken to define the more complete arctangent function for points
Complex_plane
Mathematical technique
convention is such that a positive sign is appropriate. Beam theory Bending Bending moment Singularity function Shear and moment diagram Timoshenko beam theory
Macaulay's_method
Method for assigning values to integrals
a singularity on an integral interval is avoided by limiting the integral interval to the non-singular domain. Depending on the type of singularity in
Cauchy_principal_value
Concept in algebraic topology
continuous functions as f ↦ f ∗ {\displaystyle f\mapsto f_{*}} . Here, then, C ∙ {\displaystyle C_{\bullet }} is understood to be the singular chain functor
Singular_homology
2003 science fiction novel by Charles Stross
Taliban. Singularity Sky takes place roughly in the early 23rd century, around 150 years after an event referred to by the characters as the Singularity. Shortly
Singularity_Sky
Solution to the Einstein field equations
Schwarzschild metric has a singularity for r = 0, which is an intrinsic curvature singularity. It also seems to have a singularity on the event horizon r
Schwarzschild_metric
[-1,1]} , a suitable basis choice are Bessel functions of fractional order (to capture the singularity and algebraic decay at infinity). We introduce
Fokas_method
Null hypersurface in general relativity
weak, null curvature singularity forms at the horizon. As the Cauchy horizon is approached, the gravitational field at the singularity becomes strong and
Cauchy_horizon
Singularities of holomorphic functions extend infinitely outward
More precisely, it shows that an isolated singularity is always a removable singularity for any analytic function of n > 1 complex variables. A first version
Hartogs's_extension_theorem
Theorem in complex analysis
analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes
Weierstrass factorization theorem
Weierstrass_factorization_theorem
the same on both sides of the singularity. The forms below normally assume the Cauchy principal value around a singularity in the value of C, but this is
Lists_of_integrals
American computer scientist, author and futurist (born 1948)
book, The Singularity Is Nearer: When We Merge with AI, was published in 2024. In 2010, Kurzweil wrote and co-produced the film The Singularity Is Near:
Ray_Kurzweil
Mathematical analysis of discontinuous points
distinct from an essential singularity, which is often used when studying functions of complex variables). Supposing that f is a function defined on an interval
Classification of discontinuities
Classification_of_discontinuities
Field of combinatorics using complex analysis
for a similar theorem dealing with multiple singularities. If f ( z ) {\displaystyle f(z)} has a singularity at ζ {\displaystyle \zeta } and f ( z ) ∼ (
Analytic_combinatorics
in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually
Splitting_lemma_(functions)
Ratio of polynomial functions
subring of the rational functions. The rational function f(x)=xx is equal to 1 for all x except 0, where there is a removable singularity. The sum, product
Rational_function
Mathematical functions
mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use:
Inverse_hyperbolic_functions
Operation in differential geometry
In mathematics, the jet is an operation that takes a differentiable function f and produces a polynomial, the Taylor polynomial (truncated Taylor series)
Jet_(mathematics)
Mathematical function that preserves angles
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U {\displaystyle U} and V
Conformal_map
Theoretical construct in flow physics
The Prandtl–Glauert singularity is a theoretical construct in flow physics, often incorrectly used to explain vapor cones in transonic flows. It is the
Prandtl–Glauert_singularity
Theorem in complex analysis
Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if f ( z ) {\displaystyle
Cauchy's_integral_theorem
Techniques in mathematical analysis
distinguishes whether a function or distribution is regular near x {\displaystyle x} . The information of position and covector in which a singularity occurs is encoded
Microlocal_analysis
sequences of functions. Singular function: continuous, with zero derivative almost everywhere, but non-constant. Integrable function: has an integral (finite)
List_of_types_of_functions
Riemann form Riemann function Riemann–Hurwitz formula Riemann matrix Riemann operator Riemann singularity theorem Riemann-Kempf singularity theorem Riemann
List of things named after Bernhard Riemann
List_of_things_named_after_Bernhard_Riemann
Point without a tangent space
multiplicity two and the tangent cone is not singular outside its vertex. Milnor map Resolution of singularities Singularity theory Zariski tangent space Hartshorne
Singular point of an algebraic variety
Singular_point_of_an_algebraic_variety
Special functions of several complex variables
mathematics, theta functions are special functions of several complex variables. Fundamentally, they are a family of continuous functions which encode the
Theta_function
Characteristic property of holomorphic functions
differentiability of complex functions. The equations are and where u(x, y) and v(x, y) are real bivariate differentiable functions. Typically, u and v are
Cauchy–Riemann_equations
Millennium Prize Problem
Navier–Stokes equations with a smooth external force that develops a singularity in finite time in three-dimensional Euclidean space, along with a formalization
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
the pinch point singularity one must blow-up the whole v {\displaystyle v} -axis and not only the pinch point. Whitney umbrella Singular point of an algebraic
Pinch_point_(mathematics)
Green's function for Laplacian
general nature, it is a singular integral operator, defined by convolution with a function having a mathematical singularity at the origin, the Newtonian
Newtonian_potential
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SINGULARITY FUNCTION
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SINGULARITY FUNCTION
SINGULARITY FUNCTION
SINGULARITY FUNCTION
SINGULARITY FUNCTION
SINGULARITY FUNCTION
SINGULARITY FUNCTION
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