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Generalization of the Euler gamma function and the Barnes G-function
multiple gamma function Γ N {\displaystyle \Gamma _{N}} is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function
Multiple_gamma_function
Extension of the factorial function
the gamma function, denoted by Γ {\displaystyle \Gamma } (capital Greek letter gamma), is the most common extension of the factorial function to complex
Gamma_function
Mathematical function
reciprocal gamma function is the function f ( z ) = 1 Γ ( z ) , {\displaystyle f(z)={\frac {1}{\Gamma (z)}},} where Γ(z) denotes the gamma function. Since
Reciprocal_gamma_function
Types of special mathematical functions
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to certain integrals. Their respective
Incomplete_gamma_function
Mathematical constants
The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer, half-integer, and
Particular values of the gamma function
Particular_values_of_the_gamma_function
Generalized mathematical function
In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in
Multivalued_function
Inverse of the gamma function
mathematics, the inverse gamma function Γ − 1 ( x ) {\displaystyle \Gamma ^{-1}(x)} is the inverse function of the gamma function. In other words, y = Γ
Inverse_gamma_function
Function in q-analog theory
{\displaystyle q} -gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was
Q-gamma_function
Extension of superfactorials to the complex numbers
Barnes G-function G ( z ) {\displaystyle G(z)} is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function
Barnes_G-function
Topics referred to by the same term
Γ0 or Gamma 0 may refer to: Feferman–Schütte ordinal Hecke congruence subgroup, Γ0(n) the multiple gamma function, Γn, for n = 0, as used in an inductive
Γ₀
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
Concept in mathematics
generalization of the factorial to the gamma function. There are multiple equivalent definitions of the K-function. The direct definition: K ( z ) = ( 2
K-function
Probability distribution
{\gamma (\alpha ,\beta x)}{\Gamma (\alpha )}},} where γ ( α , β x ) {\displaystyle \gamma (\alpha ,\beta x)} is the lower incomplete gamma function. If
Gamma_distribution
the multiple gamma function", Trans. Camb. Philos. Soc., 19: 374–425 Friedman, Eduardo; Ruijsenaars, Simon (2004), "Shintani–Barnes zeta and gamma functions"
Barnes_zeta_function
Probability distribution
distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution f ( x ; x 0 , γ ) {\displaystyle f(x;x_{0},\gamma )} is the distribution
Cauchy_distribution
Class of blood proteins
that gamma globulin causes the spleen to ignore the antibody-tagged platelets, thus allowing them to survive and function. Another theory on how gamma globulin
Gamma_globulin
Analytic function in mathematics
{d} x} is the gamma function. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ
Riemann_zeta_function
Probability distribution
V(x;\sigma ,\gamma )={\frac {\operatorname {Re} [w(z)]}{{\sqrt {2\pi }}\,\sigma }},} where Re[w(z)] is the real part of the Faddeeva function evaluated for
Voigt_profile
{\displaystyle \Gamma (z)} is the well known gamma function defined by Γ ( z ) = ∫ 0 ∞ t z − 1 e − z d z , ℜ ( z ) > 0 {\displaystyle \Gamma (z)=\int _{0}^{\infty
Prabhakar_function
Identity obeyed by many special functions related to the gamma function
identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values;
Multiplication_theorem
Solution of a confluent hypergeometric equation
gamma function Laguerre polynomials Parabolic cylinder function (or Weber function) Poisson–Charlier function Toronto functions Whittaker functions Mκ
Confluent hypergeometric function
Confluent_hypergeometric_function
Method of solution to differential equations
integrals of Green's functions and sums of the same. For example, if L = ( ∂ x + γ ) ( ∂ x + α ) 2 {\displaystyle L=\left(\partial _{x}+\gamma \right)\left(\partial
Green's_function
Mathematical function common in physics
_{K})^{\beta }}={\tau _{K} \over \beta }\Gamma {\left({\frac {1}{\beta }}\right)}} where Γ is the gamma function. For exponential decay, ⟨τ⟩ = τK is recovered
Stretched exponential function
Stretched_exponential_function
Mathematical concept
z ) {\displaystyle \Gamma (z+1)=z\Gamma (z)} . The Hankel contour can be used to help derive an expression for the Gamma function, based on the fundamental
Hankel_contour
Machine learning technique
we can optimize γ {\displaystyle \gamma } by finding the γ {\displaystyle \gamma } value for which the loss function has a minimum: γ m = argmin γ ∑ i
Gradient_boosting
Product of numbers from 1 to n
factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gamma function. Many other notable functions and
Factorial
Theorem in complex analysis
The theorem characterizes the gamma function, defined for x > 0 by Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t {\displaystyle \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\
Bohr–Mollerup_theorem
Term in the mathematical theory of special functions
In the mathematical theory of special functions, the Pochhammer k-symbol and the k-gamma function, introduced by Rafael Díaz and Eddy Pariguan are generalizations
Pochhammer_k-symbol
Multifractal function used in terrain modeling and simulation
weierstrass_mandelbrot_3d(x, y, D, G, L, gamma, M, n_max): """ Compute the 3D Weierstrass–Mandelbrot function z(x, y). Parameters: x, y : 2D np.ndarrays
Weierstrass–Mandelbrot function
Weierstrass–Mandelbrot_function
Generalization of gamma distribution to multiple dimensions
statistics, the Wishart distribution is a generalization of the gamma distribution to multiple dimensions. It is named in honor of John Wishart, who first
Wishart_distribution
Mathematical theorem
\int _{0}^{\infty }x^{s-1}f(x)\,dx=\Gamma (s)\,\varphi (-s)} where Γ ( s ) {\textstyle \Gamma (s)} is the gamma function. It was widely used by Ramanujan
Ramanujan's_master_theorem
α + β + γ = 180 ∘ , {\displaystyle \alpha +\beta +\gamma =180^{\circ },} as long as the functions occurring in the formulae are well-defined (the latter
List of trigonometric identities
List_of_trigonometric_identities
Integral criterion for holomorphy
_{n=1}^{\infty }{\frac {1}{n^{s}}}} or the Gamma function Γ ( α ) = ∫ 0 ∞ x α − 1 e − x d x . {\displaystyle \Gamma (\alpha )=\int _{0}^{\infty }x^{\alpha
Morera's_theorem
Fundamental trigonometric functions
the functional equation for the Gamma function, Γ ( s ) Γ ( 1 − s ) = π sin ( π s ) , {\displaystyle \Gamma (s)\Gamma (1-s)={\pi \over \sin(\pi s)},}
Sine_and_cosine
Mathematical function with multiple real-number arguments
y),\gamma (x,y))=\zeta (\alpha ,\beta ,\gamma )=e^{\alpha }[\sin(3\beta )-\cos(2\gamma )]\,.} Function composition can be used to simplify functions, which
Function of several real variables
Function_of_several_real_variables
Special mathematical function
(Vepstas 2008). Bose integral is result of multiplication between Gamma function and Zeta function. One can begin with equation for Bose integral, then use series
Polylogarithm
Economic formula of productivity
production function (in the two-factor case) is Y = A ( α L γ + ( 1 − α ) K γ ) 1 / γ , {\displaystyle Y=A\left(\alpha L^{\gamma }+(1-\alpha )K^{\gamma }\right)^{1/\gamma
Cobb–Douglas production function
Cobb–Douglas_production_function
Analytic function on the upper half-plane with a certain behavior under the modular group
the function γ ( z ) = ( a z + b ) / ( c z + d ) {\textstyle \gamma (z)=(az+b)/(cz+d)} . The identification of functions with matrices makes function composition
Modular_form
Flash of gamma rays from a distant galaxy
In gamma-ray astronomy, gamma-ray bursts (GRBs) are extremely energetic events occurring in distant galaxies that represent the brightest and most powerful
Gamma-ray_burst
Product of consecutive factorial numbers
continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function as s f ( n ) = G ( n + 2 ) {\displaystyle
Superfactorial
Probability distribution
{(k+r-1)(k+r-2)\dotsm (r)}{k!}}={\frac {\Gamma (k+r)}{k!\ \Gamma (r)}}=\left(\!\!{r \choose k}\!\!\right).} Note that Γ(r) is the Gamma function, and ( ( r k ) ) {\displaystyle
Negative binomial distribution
Negative_binomial_distribution
Graphical method of determining the stability of a dynamical system
{\displaystyle \Gamma _{s}} drawn in the complex s {\displaystyle s} plane, encompassing but not passing through any number of zeros and poles of a function F ( s
Nyquist_stability_criterion
Mathematical functions
coefficients, or any complex-valued function. Falling factorials appear in multiple differentiation of simple power functions: ( d d x ) n x a = ( a ) n ⋅ x
Falling_and_rising_factorials
Mathematical function
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as
Jacobi_elliptic_functions
Neural oscillation in the 25–140Hz range
A gamma wave or gamma rhythm is a pattern of neural oscillation in humans with a frequency between 30 and 100 Hz, the 40 Hz point being of particular
Gamma_wave
Model for coordination and decision-making among multiple agents
, { Ω i } , O , γ ) {\displaystyle (S,\{A_{i}\},T,R,\{\Omega _{i}\},O,\gamma )} , where S {\displaystyle S} is a set of states, A i {\displaystyle A_{i}}
Decentralized partially observable Markov decision process
Decentralized_partially_observable_Markov_decision_process
Number of integers coprime to and less than n
{\displaystyle \gamma } is Euler's constant and p 120569 # {\displaystyle p_{120569}\#} is the product of the first 120569 primes. Carmichael function (λ) Dedekind
Euler's_totient_function
Probability distribution
is the number of degrees of freedom, and Γ {\displaystyle \Gamma } is the gamma function. This may also be written as f ( t ) = 1 ν B ( 1 2 , ν 2 ) (
Student's_t-distribution
Generalization of the hypergeometric function
(1-b_{j}+s)\prod _{j=n+1}^{p}\Gamma (a_{j}-s)}}\,z^{s}\,ds,} where Γ denotes the gamma function. This integral is of the so-called Mellin–Barnes type, and may be viewed
Meijer_G-function
Rules for computing derivatives of functions
{1}{x+n}}\right)-{\dfrac {1}{x}}\right)\\&=\Gamma (x)\psi (x),\end{aligned}}} with ψ ( x ) {\textstyle \psi (x)} being the digamma function, expressed by the parenthesized
Differentiation_rules
Function in fluid dynamics
The J-function is defined as: J ( S w ) = p c ( S w ) k / ϕ γ cos θ {\displaystyle J(S_{w})={\frac {p_{c}(S_{w}){\sqrt {k/\phi }}}{\gamma \cos \theta
Leverett_J-function
Cancer of plasma cells
kidney function may develop, either acutely or chronically, and with any degree of severity. The most common cause of kidney failure in multiple myeloma
Multiple_myeloma
Probability distribution
It is a special case of the inverse-gamma distribution and a stable distribution. The probability density function of the Lévy distribution over the domain
Lévy_distribution
Type of mathematical function
{\displaystyle \gamma } such that α < γ {\displaystyle \alpha <\gamma } and β < γ {\displaystyle \beta <\gamma } and γ {\displaystyle \gamma } is not the
Ordinal_notation
Class of reinforcement learning algorithms
S_{t})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau }){\Big |}S_{0}=s_{0}\right]} Lemma—The expectation of the score function is zero, conditional on any
Policy_gradient_method
Time series statistical test
function ndiffs handles multiple popular unit root tests package tseries function adf.test package fUnitRoots function adfTest package urca function ur
Augmented_Dickey–Fuller_test
Two-dimensional conformal field theory
iP_{2}\pm iP_{3})}}\ ,} where the special function Υ b {\displaystyle \Upsilon _{b}} is a kind of multiple gamma function. For c ∈ ( − ∞ , 1 ) {\displaystyle
Liouville_field_theory
Mathematical method in calculus
\end{aligned}}} may be derived using integration by parts. The gamma function is an example of a special function, defined as an improper integral for z > 0 {\displaystyle
Integration_by_parts
Enzyme
Gamma-glutamyl carboxylase is an enzyme that in humans is encoded by the GGCX gene, located on chromosome 2 at 2p12. Gamma-glutamyl carboxylase is an enzyme
Gamma-glutamyl_carboxylase
Function that is holomorphic on the whole complex plane
sigma function. Other examples include the Fresnel integrals, the Jacobi theta function, and the reciprocal Gamma function. The exponential function and
Entire_function
Mathematical function
everywhere it is defined. As with the gamma function that extends the ordinary factorial function, this double factorial function is logarithmically convex in
Double_factorial
Field of combinatorics using complex analysis
{n^{-\alpha -1}}{\Gamma (-\alpha )}}(\log n)^{\gamma }(\log \log n)^{\delta }\quad } as n → ∞ {\displaystyle n\to \infty } For generating functions including
Analytic_combinatorics
Function in analytic number theory
positive real part ( Γ ( s ) {\displaystyle \Gamma (s)} represents the gamma function). This gives the eta function as a Mellin transform. Hardy gave a simple
Dirichlet_eta_function
Locally defined function in general relativity
′ {\displaystyle \gamma (\lambda _{0})=x'} and γ ( λ 1 ) = x {\displaystyle \gamma (\lambda _{1})=x} . Then Synge's world function is defined as: σ (
Synge's_world_function
f_{\text{R}}(q;k,\nu )={\frac {{\sqrt {2\pi \,}}\,k\,(k-1)\,\nu ^{\nu /2}}{\Gamma (\nu /2)\,2^{\left(\nu /2-1\right)}}}\int _{0}^{\infty }s^{\nu }\,\varphi
Studentized range distribution
Studentized_range_distribution
Theorem on prime numbers
have practical value. Wilson's theorem allows one to define the p-adic gamma function. Gauss proved that ∏ k = 1 gcd ( k , m ) = 1 m − 1 k ≡ { − 1 ( mod
Wilson's_theorem
Family of continuous probability distributions
{\gamma (k,\lambda x)}{\Gamma (k)}}={\frac {\gamma (k,\lambda x)}{(k-1)!}},} where γ {\displaystyle \gamma } is the lower incomplete gamma function and
Erlang_distribution
Method of interpolation
( h ) . {\displaystyle \gamma {\big (}Z(x_{1}),Z(x_{2}){\big )}=\gamma {\big (}Z(x_{i}),Z(x_{i}+\mathbf {h} ){\big )}=\gamma (h).} For simplicity, we
Kriging
Formulation of classical mechanics
{\displaystyle \gamma =\gamma (\tau ;t,t_{0},\mathbf {q} ,\mathbf {q} _{0})} be the (unique) extremal from the definition of the Hamilton's principal function S
Hamilton–Jacobi_equation
Statistical distribution
}}x^{2}\right)}{\Gamma (m)}}=P\left(m,{\frac {m}{\Omega }}x^{2}\right)} where P is the regularized (lower) incomplete gamma function. The parameters m
Nakagami_distribution
Diagnostic imaging test in nuclear medicine
tomography (PET) also collect 2-D images using gamma cameras, but they collect multiple 2-D images at multiple angles, from which they reconstruct 3-dimensional
Scintigraphy
Discrete probability distribution
using the lgamma function in the C standard library (C99 version) or R, the gammaln function in MATLAB or SciPy, or the log_gamma function in Fortran 2008
Poisson_distribution
Probability distribution
} where W is the Whittaker function while β = n 1 − ρ , γ = n 1 + ρ {\displaystyle \beta ={\frac {n}{1-\rho }},\;\;\gamma ={\frac {n}{1+\rho }}} . Using
Distribution of the product of two random variables
Distribution_of_the_product_of_two_random_variables
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
Multiple comparison procedure
α p = 1 − γ p {\displaystyle \alpha _{p}=1-\gamma _{p}} , γ p = ( 1 − α ) ( p − 1 ) {\displaystyle \gamma _{p}=(1-\alpha )^{(p-1)}} and p {\displaystyle
Duncan's new multiple range test
Duncan's_new_multiple_range_test
Number of subsets of a given size
generalized to two real or complex valued arguments using the gamma function or beta function via ( x y ) = Γ ( x + 1 ) Γ ( y + 1 ) Γ ( x − y + 1 ) = 1 (
Binomial_coefficient
Computer programming concept
methods which learn by bootstrapping from the current estimate of the value function. These methods sample from the environment, like Monte Carlo methods, and
Temporal_difference_learning
Chemical compound
Kemmel V, Andriamampandry C, Maitre M (December 1999). "Gamma-hydroxybutyrate receptor function studied by the modulation of nitric oxide synthase activity
Γ-Hydroxybutyric_acid
Probability distribution
normalizing constant is the multivariate beta function, which can be expressed in terms of the gamma function: B ( α ) = ∏ i = 1 K Γ ( α i ) Γ ( ∑ i = 1
Dirichlet_distribution
Multivalued function in mathematics
{1}{N}}}\Gamma \left(1-{\frac {1}{N}}\right)\qquad {\text{for }}N>1\end{aligned}}} where Γ {\displaystyle \Gamma } denotes the gamma function. The first
Lambert_W_function
Concept in mathematics
Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be
Walsh_function
Size of a mathematical ball
recurrence relation. Closed-form expressions involve the gamma, factorial, or double factorial function. The volume can also be expressed in terms of A n {\displaystyle
Volume_of_an_n-ball
Special mathematical function defined as sin(x)/x
}\left(1-{\frac {x^{2}}{n^{2}}}\right)} and is related to the gamma function Γ(x), as well as to Gauss' Pi function, through Euler's reflection formula: sin ( π x
Sinc_function
Ability of certain substances to produce several distinct biological responses
so they can be attacked, among other functions. TH1 cells are created to make cytokines, like interferon gamma, that activate macrophages and cytotoxic
Pluripotency (biological compounds)
Pluripotency_(biological_compounds)
Probability distribution
-1}\end{aligned}}} where Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. The beta function, B {\displaystyle \mathrm {B} } , is a normalization
Beta_distribution
Standard for video display gamma
ITU-R BT.1886 is the reference EOTF of SDR-TV. It is a gamma 2.4 transfer function (a power law with a 2.4 exponent) considered as a satisfactory approximation
ITU-R_BT.1886
Numbers expressible as integrals of algebraic functions
a complex number that can be expressed as an integral of an algebraic function over an algebraic domain. The periods are a class of numbers which includes
Period_(number_theory)
Formal power series
{\beta }{n}}\!\!\right)\left({\frac {1}{r}}\right)^{n}\,,} using the gamma function, a binomial coefficient, or a multiset coefficient. Note that limit
Generating_function
Conjecture on zeros of the zeta function
(n)}}<e^{\gamma }\log \log n+{\frac {e^{\gamma }(4+\gamma -\log 4\pi )}{\sqrt {\log n}}}} is true for all n ≥ 120569#, where φ(n) is Euler's totient function and
Riemann_hypothesis
Class of statistical models
canonical link functions and their inverses (sometimes referred to as the mean function, as done here). In the cases of the exponential and gamma distributions
Generalized_linear_model
Branch of mathematical analysis
a generalization for real n: using the gamma function to remove the discrete nature of the factorial function gives us a natural candidate for applications
Fractional_calculus
Family of probability distributions
continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions
Tweedie_distribution
Mathematical model for sequential decision making under uncertainty
V^{*}(s)=\max _{a}E\left[R_{a}(s,s')+\gamma V^{*}(s')\right]} From inspection, notice that this fixed point is the value function associated to the following policy
Markov_decision_process
Total species diversity in a landscape
up a concept gone awry. Part 1. Defining beta diversity as a function of alpha and gamma diversity. Ecography 33: 2-22. doi:10.1111/j.1600-0587.2009.05880
Gamma_diversity
Technique to make a model more generalizable and transferable
{\gamma }{n}}{\hat {X}}^{\mathsf {T}}{\hat {X}}\right){\frac {\gamma }{n}}\sum _{i=0}^{T-2}\left(I-{\frac {\gamma }{n}}{\hat {X}}^{\mathsf
Regularization_(mathematics)
C library for arbitrary-precision floating-point arithmetic
exp(x)−1 functions (log1p and expm1), the six trigonometric and hyperbolic functions and their inverses, the gamma, zeta and error functions, the arithmetic–geometric
GNU_MPFR
Type of generalization of periodic functions in Euclidean space
< G {\displaystyle \Gamma <G} of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to
Automorphic_form
Graphical aid for deriving some concepts in combinatorics
objects are not distinguished). This is represented by the generating function 1 + 1 x + 1 x 2 + 1 x 3 + … = 1 + x + x 2 + x 3 + … = 1 1 − x . {\displaystyle
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Nearest integers from a number
Floor and ceiling functions In mathematics, the floor function is the function that takes a real number x as input and returns the greatest integer less
Floor_and_ceiling_functions
Sequence of differential equation solutions
}}\Re (\gamma )>-{\tfrac {1}{2}}} for the exponential function. The incomplete gamma function has the representation Γ ( α , x ) = x α e − x ∑ i = 0
Laguerre_polynomials
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MULTIPLE GAMMA-FUNCTION
MULTIPLE GAMMA-FUNCTION
MULTIPLE GAMMA-FUNCTION
MULTIPLE GAMMA-FUNCTION
MULTIPLE GAMMA-FUNCTION
MULTIPLE GAMMA-FUNCTION
MULTIPLE GAMMA-FUNCTION
MULTIPLE GAMMA-FUNCTION
MULTIPLE GAMMA-FUNCTION
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