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Topics referred to by the same term
G-function, related to the Gamma function Meijer G-function, a generalization of the hypergeometric function Siegel G-function, a class of functions in
G-function
Extension of superfactorials to the complex numbers
In mathematics, the Barnes G-function G ( z ) {\displaystyle G(z)} is a function that is an extension of superfactorials to the complex numbers. It is
Barnes_G-function
Generalization of the hypergeometric function
the G-function was introduced by Cornelis Simon Meijer (1936) as a very general function intended to include most of the known special functions as particular
Meijer_G-function
Association of one output to each input
possible applications of the concept. A function is often denoted by a letter such as f, g or h. The value of a function f at an element x of its domain (that
Function_(mathematics)
Class of functions in transcendental number theory
In mathematics, the Siegel G-functions are a class of functions in transcendental number theory introduced by C. L. Siegel. They satisfy a linear differential
Siegel_G-function
Extension of the factorial function
function g ( x ) {\displaystyle g(x)} with g ( x ) = g ( x + 1 ) {\displaystyle g(x)=g(x+1)} and g ( 0 ) = 1 {\displaystyle g(0)=1} , such as g (
Gamma_function
Degree of differentiability of a function or map
but not of class Cj where j > k. The function g(x)={x2sin(1x)if x≠0,0if x=0 is differentiable, with derivative g′(x)={−cos(1x)+2xsin(1x)if x≠0,0if x=0
Smoothness
Method of solution to differential equations
linear differential operator, then the Green's function G {\displaystyle G} is the solution of the equation L G = δ , {\displaystyle LG=\delta ,} where δ {\displaystyle
Green's_function
Operation on mathematical functions
two functions, f {\displaystyle f} and g {\displaystyle g} , and returns a new function f ∘ g {\displaystyle f\circ g} . When the composite function f ∘
Function_composition
Theoretical framework in harmonic analysis
involves studying a function by decomposing it in terms of functions with localized frequencies, and using the Littlewood–Paley g-function to compare it with
Littlewood–Paley_theory
Generalization of the Euler gamma function and the Barnes G-function
gamma function Γ N {\displaystyle \Gamma _{N}} is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was
Multiple_gamma_function
Generalized function whose value is zero everywhere except at zero
and g {\displaystyle g} are functions such that f = g {\displaystyle f=g} almost everywhere, then f {\displaystyle f} is integrable if and only if g {\displaystyle
Dirac_delta_function
Description of continuous random distribution
probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function whose value at any given
Probability_density_function
Mathematical concept
exists a function g from Y to X such that g(f(x))=x for all x∈X and f(g(y))=y for all y∈Y. If f is invertible, then there is exactly one function g satisfying
Inverse_function
Product of consecutive factorial numbers
gamma function, the superfactorials can be continuously interpolated by the Barnes G-function as s f ( n ) = G ( n + 2 ) {\displaystyle sf(n)=G(n+2)}
Superfactorial
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
Transcendental single-variable function
tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function. The Clausen function of order 2 – often referred
Clausen_function
Mathematical function such that every output has at least one input
words, g is a right inverse of f if the composition f o g of g and f in that order is the identity function on the domain Y of g. The function g need not
Surjective_function
Generalization of the Meijer G-function and the Fox–Wright function
mathematics, the Fox H-function H ( z ) {\displaystyle H(z)} is a generalization of the Meijer G-function and the Fox–Wright function introduced by Charles
Fox_H-function
Integral expressing the amount of overlap of one function as it is shifted over another
mathematical operation on two functions f and g that produces a third function f∗g, as the integral of the product of the two functions after one is reflected
Convolution
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Function that is holomorphic on the whole complex plane
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane
Entire_function
types of functions Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...) Algebraic functions are functions
List of mathematical functions
List_of_mathematical_functions
Continuous function that is not absolutely continuous
Function composition extends this to a monoid, in that one can write g 010 = g 0 g 1 g 0 {\displaystyle g_{010}=g_{0}g_{1}g_{0}} and generally, g A g
Cantor_function
Evaluation of a function on its argument
example: f(g(h(j(x)))) can be rewritten as: f$g$h$j$x This can be equivalently expressed using function composition as: (f∘g∘h∘j)(x) or even: (f∘g∘h∘j∘x)()
Function_application
Real function with secant line between points above the graph itself
function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function
Convex_function
Order-preserving mathematical function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept
Monotonic_function
Fast-growing function
Friedman's SSCG function is a mathematical function defined by Harvey Friedman. It is defined by SSCG ( k ) {\displaystyle {\text{SSCG}}(k)} as the largest
Friedman's_SSCG_function
class function is a function on a group G that is constant on the conjugacy classes of G. In other words, it is invariant under the conjugation map on G. Such
Class_function
Type of regression analysis
into a moments problem in a natural function space, usually built around generalizations of the Meijer-G function. By not requiring a priori specification
Symbolic_regression
define the Ramanujan G- and g-functions as 2 1 / 4 G n = q − 1 24 ∏ n > 0 ( 1 + q 2 n − 1 ) = η 2 ( τ ) η ( τ 2 ) η ( 2 τ ) , 2 1 / 4 g n = q − 1 24 ∏ n >
Weber_modular_function
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Mathematical function with no sudden changes
In mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies
Continuous_function
Mathematical formula
G in terms of a maximal torus T. Precisely, it says there exists a real-valued continuous function u on T such that for every class function f on G (function
Weyl_integration_formula
Concept in complex analysis
function g is a function whose complex derivative is g. More precisely, given an open set U {\displaystyle U} in the complex plane and a function g :
Antiderivative (complex analysis)
Antiderivative_(complex_analysis)
Function specifying the behavior of a component in an electronic or control system
a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that models
Transfer_function
Mathematical function
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial
Beta_function
Function with a multiplicative scaling behaviour
mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by
Homogeneous_function
Function that is discontinuous at rationals and continuous at irrationals
Thomae's function is a real-valued function of a real variable that can be defined as: f ( x ) = { 1 q if x = p q ( x is rational), with p ∈ Z and
Thomae's_function
Mathematical transform that expresses a function of time as a function of frequency
exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced sine
Fourier_transform
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Mathematical constant
is a mathematical constant, related to special functions like the K-function and the Barnes G-function. The constant also appears in a number of sums
Glaisher–Kinkelin_constant
Indefinite integral
and G. Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over
Antiderivative
always be expressed in terms of the Meijer G-function, while the opposite is not true, so that the G-function is of a still more general nature. It is defined
MacRobert_E_function
G {\displaystyle L:{\mathcal {F}}\to {\mathcal {G}}} which takes a function y ∈ F {\displaystyle y\in {\mathcal {F}}} to another function L [ y ] ∈ G
List_of_mathematic_operators
function such that g ( 0 ) = 0 {\displaystyle g(0)=0} and g ( 1 ) = 1 {\displaystyle g(1)=1} . The dual distortion function is g ~ ( x ) = 1 − g ( 1 − x ) {\displaystyle
Distortion_function
interest in transcendental number theory, and are closely related to G-functions. A power series with coefficients in the field of algebraic numbers f
E-function
composition of an univariate function g : R → R {\displaystyle g:\mathbb {R} \rightarrow \mathbb {R} } , that is called a profile function, with an affine transformation
Ridge_function
Real function with finite total variation
and to the y-axis. A function g {\textstyle g} has bounded variation if and only if the Riemann–Stieltjes integral ∫ a b f d g {\textstyle \int _{a}^{b}f\
Bounded_variation
Mathematical rule for evaluating limits
the Swiss mathematician Johann Bernoulli. For two functions f {\displaystyle f} and g {\displaystyle g} , under most circumstances the limit of their quotient
L'Hôpital's_rule
Mathematical function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f ( x ) = exp ( − x 2 ) {\displaystyle f(x)=\exp(-x^{2})}
Gaussian_function
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Function that is continuous everywhere but differentiable nowhere
the cosine function can be replaced in the infinite series by a piecewise linear "zigzag" function. G. H. Hardy showed that the function of the above
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
Function that takes one or more functions as an input or that outputs a function
plusthree←+∘3 g←plusthree twice g 7 13 Using std::function<R(Args...)> in C++11: import std; using std::function; auto twice = [](const function<int(int)>&
Higher-order_function
Formal power series
generating function and its integral: G ′ ( z ) = ∑ n = 0 ∞ ( n + 1 ) g n + 1 z n z ⋅ G ′ ( z ) = ∑ n = 0 ∞ n g n z n ∫ 0 z G ( t ) d t = ∑ n = 1 ∞ g n − 1
Generating_function
Smooth and compactly supported function
d, the function R ∋ x ↦ g ( x − a b − a ) g ( d − x d − c ) {\displaystyle \mathbb {R} \ni x\mapsto g{\Bigl (}{\frac {x-a}{b-a}}{\Bigr )}\,g{\Bigl (}{\frac
Bump_function
Polynomial function of degree two
In mathematics, a quadratic function of a single variable is a function of the form f ( x ) = a x 2 + b x + c {\displaystyle f(x)=ax^{2}+bx+c} with
Quadratic_function
Functions of an angle
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of
Trigonometric_functions
Negative of a convex function
a concave function is also a global maximum. A strictly concave function will have at most one global maximum. The functions f(x)=−x2 and g(x)=x are concave
Concave_function
Point to which functions converge in analysis
and g are real-valued (or complex-valued) functions, then taking the limit of an operation on f(x) and g(x) (e.g., f + g, f − g, f × g, f / g, f g) under
Limit_of_a_function
Quickly growing function
version of the Ackermann function that became preferred by almost all authors. The generalized hyperoperation sequence, e.g. G(m,a,b)=a[m]b, is a version
Ackermann_function
Covariance and correlation
where K g = [ k ( g , T 0 ( g ) ) , k ( g , T 1 ( g ) ) , … , k ( g , T N − 1 ( g ) ) ] {\displaystyle K_{g}=[k(g,T_{0}(g)),k(g,T_{1}(g)),\dots ,k(g,T_{N-1}(g))]}
Cross-correlation
Theorem in mathematics
mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that if
Inverse_function_theorem
Property of operations
( g ∘ g ) = f ∘ g {\displaystyle (f\circ g)\circ (f\circ g)=f\circ (g\circ f)\circ g=f\circ (f\circ g)\circ g=(f\circ f)\circ (g\circ g)=f\circ g} ,
Idempotence
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized
Rectangular_function
Type of function
functions over the interval: ⟨ f , g ⟩ = ∫ f ( x ) ¯ g ( x ) d x . {\displaystyle \langle f,g\rangle =\int {\overline {f(x)}}g(x)\,dx.} The functions
Orthogonal_functions
Class of statistical models
and 3. A link function g {\displaystyle g} such that E ( Y ∣ X ) = μ = g − 1 ( η ) {\displaystyle \operatorname {E} (Y\mid X)=\mu =g^{-1}(\eta )} .
Generalized_linear_model
Block cipher
Feistel network with an F-function operating on 64-bit halves, while the F-function itself is a Feistel network composed of a G-function operating on 32-bit
SEED
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
S-shaped curve
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation f ( x ) = L 1 + e − k ( x − x 0 ) {\displaystyle f(x)={\frac
Logistic_function
Reason for a change under natural selection; in physiology, what a system does
In evolutionary biology, function is the reason some object or process occurred in a system that evolved through natural selection. That reason is typically
Function_(biology)
Representation of a mathematical function
set Y (the codomain), the graph of the function is the set G ( f ) = { ( x , f ( x ) ) : x ∈ X } , {\displaystyle G(f)=\{(x,f(x)):x\in X\},} which is a subset
Graph_of_a_function
Class of mathematical functions
{ − ∞ } {\displaystyle \varphi \colon G\to \mathbb {R} \cup \{-\infty \}} be an upper semi-continuous function. Then, φ {\displaystyle \varphi } is called
Subharmonic_function
Askey–Wilson polynomial, Askey–Wilson function (with James A. Wilson) Ernest William Barnes: Barnes G-function E. T. Bell: Bell polynomials Bender–Dunne
List of eponyms of special functions
List_of_eponyms_of_special_functions
Approximation of a function by a polynomial
to the function g(t)=f(𝒖(t)): f(𝒙)=g(1)=g(0)+∑j=1k1j!g(j)(0) + ∫01(1−t)kk!g(k+1)(t)dt. Applying the chain rule for several variables gives g
Taylor's_theorem
Artificial neural network node function
In artificial neural networks, the activation function of a node is a function that calculates the output of the node based on its individual inputs and
Activation_function
Concept in complexity theory
every computable function f {\displaystyle f} , there is a computable function g {\displaystyle g} that is time constructible and ∀ n , g ( n ) > f ( n )
Constructible_function
Mathematical relation consisting of a multi-variable function equal to zero
unique inverse function. If g is a function of x that has a unique inverse, then the inverse function of g, called g−1, is the unique function giving a solution
Implicit_function
Functions such that f(–x) equals f(x) or –f(x)
In mathematics, an even function is a real function such that f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} for every x {\displaystyle x} in its domain
Even_and_odd_functions
Dutch mathematician (1904–1974)
introduced the Meijer G-function, a very general function that includes most of the elementary and higher mathematical functions as special cases; he also
Cornelis_Simon_Meijer
Topics referred to by the same term
various G {\displaystyle G} -functions g {\displaystyle g} , Hedges' g, a measure of effect size Gauss (unit), a unit of magnetic induction G, electrical
G_(disambiguation)
use objects, for instance f(x) instead of x.f(), or g(x, y) instead of x.g(y). Friend functions have the same implications on encapsulation as methods
Friend_function
Matrix of second derivatives
function f considered previously, but adding a constraint function g such that g(𝐱)=c, the bordered Hessian is the Hessian of the Lagrange function Λ(𝐱
Hessian_matrix
Product of numbers from 1 to n
factorials. The superfactorials are continuously interpolated by the Barnes G-function. Triangular number Just as the n {\displaystyle n} th factorial is the
Factorial
Mathematical function characterizing set membership
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all
Indicator_function
Function used in signal processing
processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside
Window_function
Functions in mathematics
the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f\colon U\to \mathbb
Harmonic_function
Analytic function in mathematics
Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ {\displaystyle \zeta } (zeta), is a function of a complex
Riemann_zeta_function
Class of mathematical functions
elliptic functions with respect to a given period lattice. A cubic of the form C g 2 , g 3 C = { ( x , y ) ∈ C 2 : y 2 = 4 x 3 − g 2 x − g 3 } {\displaystyle
Weierstrass_elliptic_function
Function whose actual domain of definition may be smaller than its apparent domain
In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that
Partial_function
Rules for computing derivatives of functions
concise way as: [D(f∘g)]x=[Df]g(x)⋅[Dg]x. If the function f has an inverse function g, meaning that g(f(x))=x and f(g(y))=y, then: g′=1f′∘g. In Leibniz notation
Differentiation_rules
Class of mathematical function
meromorphic functions can be added, subtracted, multiplied, and the quotient f / g {\displaystyle f/g} can be formed unless g ( z ) = 0 {\displaystyle g(z)=0}
Meromorphic_function
Mathematical definition of point elasticity
mathematics, the elasticity or point elasticity of a positive differentiable function f of a positive variable (positive input, positive output) at point a is
Elasticity_of_a_function
One-to-one correspondence
there is a function g : Y → X , {\displaystyle g:Y\to X,} the inverse of f, such that each of the two ways for composing the two functions produces an
Bijection
Mathematical constants
Barnes G-function: Γ ( i ) = G ( 1 + i ) G ( i ) = e − log G ( i ) + log G ( 1 + i ) . {\displaystyle \Gamma (i)={\frac {G(1+i)}{G(i)}}=e^{-\log G(i)+\log
Particular values of the gamma function
Particular_values_of_the_gamma_function
Special function in mathematics
In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1
Hurwitz_zeta_function
Concept in dynamical systems
would end up with a function g {\displaystyle g} that satisfies g ( x ) = − α g ( g ( − x / α ) ) {\displaystyle g(x)=-\alpha g(g(-x/\alpha ))} . Further
Feigenbaum_function
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
In functional programming
application (or partial function application) refers to the process of fixing a number of arguments of a function, producing another function of smaller arity
Partial_application
Function that, applied twice, gives another function
root of a function with respect to the operation of function composition. In other words, a functional square root of a function g is a function f satisfying
Functional_square_root
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