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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
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
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
Third letter of the Greek alphabet
the gamma function (usually written as Γ-function) is an extension of the factorial to complex numbers In mathematics, the upper incomplete gamma function
Gamma
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
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
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
Extension of the factorial function
Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function (it is an
Hadamard's_gamma_function
Two-parameter family of continuous probability distributions
scaled inverse chi-squared distribution. The inverse gamma distribution's probability density function is defined over the support x > 0 {\displaystyle x>0}
Inverse-gamma_distribution
Difference between logarithm and harmonic series
for the gamma function and the Barnes G-function. The asymptotic expansion of the gamma function, Γ ( 1 / x ) ∼ x − γ {\displaystyle \Gamma (1/x)\sim
Euler's_constant
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
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
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
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
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
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
Mathematical function
In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by Morita
P-adic_gamma_function
Probability distribution
Gamma (d/p)}},} where Γ ( ⋅ ) {\displaystyle \Gamma (\cdot )} denotes the gamma function. The cumulative distribution function is F ( x ; a
Generalized gamma distribution
Generalized_gamma_distribution
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
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
Image luminance mapping function
Gamma correction or gamma is a nonlinear operation used to encode and decode luminance in video or images. Gamma correction is, in the simplest cases,
Gamma_correction
Multivariate generalization of the gamma function
gamma function Γp is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of
Multivariate_gamma_function
Meromorphic function
\mathbb {C} } defined as the (m + 1)th derivative of the logarithm of the gamma function: ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln Γ ( z )
Polygamma_function
Probability distribution
moment-generating function is MX(t)=𝔼[etX]=(1−p+pet)n. Usually the mode of a binomial B(n, p) distribution is equal to ⌊(n+1)p⌋, where ⌊⋅⌋ is the floor function. However
Binomial_distribution
function, Polygamma function Incomplete beta function Incomplete gamma function K-function Multivariate gamma function: A generalization of the Gamma
List of mathematical functions
List_of_mathematical_functions
Number, approximately 3.14
gamma function is evaluated at half-integers, the result is naturally expressed in terms of π. For example, Γ(12)=π and Γ(52)=34π. The gamma function
Pi
Meromorphic function on the complex plane
{s+\kappa _{j}}{2}}\right)} where Γ {\displaystyle \textstyle \Gamma } denotes the gamma function, π {\displaystyle \textstyle \pi } denotes the automorphic
L-function
Analytic function that does not satisfy a polynomial equation
logarithm and inverse trigonometric functions. All special functions such as the gamma, error, bessel, and Riemann zeta functions are transcendental. Equations
Transcendental_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
Mathematical function
_{k=0}^{\infty }{\frac {z^{k}}{\Gamma (\alpha k+1)}},} where Γ ( x ) {\displaystyle \Gamma (x)} is the gamma function, and α {\displaystyle \alpha } is
Mittag-Leffler_function
Special functions of several complex variables
Many values of the theta function and especially of the shown phi function can be represented in terms of the gamma function: φ ( exp ( − 2 π ) ) =
Theta_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
Mathematic function
the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely
Elliptic_gamma_function
Conventional name of the gamma distribution when applied to macromolecular polydispersity
function is f ( x ) = k k x k − 1 e − k x Γ ( k ) , {\displaystyle f(x)={\frac {k^{k}x^{k-1}e^{-kx}}{\Gamma (k)}},} where Γ(x) is the Gamma function.
Schulz–Zimm_distribution
Numerical method for calculating the gamma function
mathematics, the Lanczos approximation is a method for computing the gamma function numerically, published by Cornelius Lanczos in 1964. It is a practical
Lanczos_approximation
Special function in the physical sciences
{1}{3}}\right)}}.\end{aligned}}} Here, Γ {\displaystyle \Gamma } denotes the gamma function. It follows that the Wronskian of Ai ( x ) {\displaystyle
Airy_function
Function that interpolates the factorial
In mathematics, a pseudogamma function is a function that interpolates the factorial. The gamma function is the most famous solution to the problem of
Pseudogamma_function
Type of permutation of a set of elements
Dn=Γ(n+1,−1)e=∫0∞(x−1)ne−xdx where Γ(s, x) is the upper incomplete gamma function. It is NP-complete to determine whether a given permutation group (described
Derangement
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
Probability distribution
Gamma \left({\frac {k}{2}}\right)}},&x\geq 0;\\0,&{\text{otherwise}}.\end{cases}}} where Γ ( z ) {\displaystyle \Gamma (z)} is the gamma function. The
Chi_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
Function defined by a hypergeometric series
non-negative integer, one has 2F1(z) → ∞. Dividing by the value Γ(c) of the gamma function, we have the limit: lim c → − m 2 F 1 ( a , b ; c ; z ) Γ ( c ) = (
Hypergeometric_function
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
Number computed as a product of powers
continuously interpolated by the gamma function, the hyperfactorials can be continuously interpolated by the K-function as K ( n + 1 ) = H ( n ) {\displaystyle
Hyperfactorial
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
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
Evaluates a certain product of values of the Gamma function at rational values
certain product of values of the gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadratic irrational
Chowla–Selberg_formula
Family of solutions to related differential equations
_{m=0}^{\infty }{\frac {(-1)^{m}}{m!\,\Gamma (m+\alpha +1)}}{\left({\frac {x}{2}}\right)}^{2m+\alpha },} where Γ(z) is the gamma function, a shifted generalization
Bessel_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
Class of mathematical function
{z}}{(z-1)^{2}}}} as well as the gamma function and the Riemann zeta function are meromorphic on the whole complex plane. The function f ( z ) = e 1 z {\displaystyle
Meromorphic_function
Graphical aid for deriving some concepts in combinatorics
objects are not distinguished). This is represented by the generating function 1+1x+1x2+1x3+…=1+x+x2+x3+…=11−x. The series is a geometric series, and
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Approximation for factorials
= Γ ( n + 1 ) , {\displaystyle n!=\Gamma (n+1),} where Γ denotes the gamma function. However, the gamma function, unlike the factorial, is more broadly
Stirling's_approximation
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
Fundamental trigonometric functions
equation for the Gamma function, Γ(s)Γ(1−s)=πsin(πs), which in turn is found in the functional equation for the Riemann zeta-function,
Sine_and_cosine
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
Stochastic process for effort or wear
Γ ( ⋅ ) {\displaystyle \Gamma (\cdot )} (the gamma function), Γ ( γ , λ ) {\displaystyle \Gamma (\gamma ,\lambda )} (the gamma distribution), and Γ ( t
Gamma_process
Mathematical function
function is defined in terms of the gamma function as θ ( t ) = arg ( Γ ( 1 4 + i t 2 ) ) − log π 2 t {\displaystyle \theta (t)=\arg \left(\Gamma
Riemann–Siegel_theta_function
Triangular array of the binomial coefficients
convolving the distribution function for a random variable with itself corresponds to calculating the distribution function for a sum of n independent
Pascal's_triangle
Family of power series in mathematics
(a)_{n}=a(a+1)(a+2)\cdots (a+n-1)={\frac {\Gamma (a+n)}{\Gamma (a)}}} where Γ {\displaystyle \Gamma } represents the gamma function. The series can then be written
Generalized hypergeometric function
Generalized_hypergeometric_function
mathematics, Gautschi's inequality is an inequality for ratios of gamma functions. It is named after Walter Gautschi. Let x {\displaystyle x} be a positive
Gautschi's_inequality
Special function in mathematics
zeta function has an integral representation ζ ( s , a ) = 1 Γ ( s ) ∫ 0 ∞ x s − 1 e − a x 1 − e − x d x {\displaystyle \zeta (s,a)={\frac {1}{\Gamma (s)}}\int
Hurwitz_zeta_function
Sigmoid shape special function
[further explanation needed] In terms of the regularized gamma function P and the incomplete gamma function, erf ( x ) = sgn ( x ) ⋅ P ( 1 2 , x 2 ) = sgn
Error_function
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
Numerical method for calculating the gamma function
Spouge's approximation is a formula for computing an approximation of the gamma function. It was named after John L. Spouge, who defined the formula in a 1994
Spouge's_approximation
Integral of the Gaussian function, equal to sqrt(π)
t {\textstyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}dt} is the gamma function. More generally, ∫ 0 ∞ x n e − a x b d x = Γ ( ( n + 1 ) / b ) b a (
Gaussian_integral
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
Result on gamma function
theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first
Hölder's_theorem
Algebraic expansion of powers of a binomial
{n}{k}}={\frac {n!}{k!\;(n-k)!}},} which is defined in terms of the factorial function n!. Equivalently, this formula can be written ( n k ) = n ( n − 1 ) ⋯ (
Binomial_theorem
Mathematical series
to Euler's definition of the Gamma function: Γ ( z ) = lim k → ∞ k ! k z z ( z + 1 ) ⋯ ( z + k ) , {\displaystyle \Gamma (z)=\lim _{k\to \infty }{\frac
Binomial_series
Symbols for constants, special functions
optical mode in a waveguide the gamma function, a generalization of the factorial the upper incomplete gamma function the modular group, the group of
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
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
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
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
Mathematical operation
transform and to the theory of the gamma function and similar special functions. The Mellin transform of a complex-valued function f {\displaystyle f} defined
Mellin_transform
Topics referred to by the same term
{\displaystyle \Pi (x)\,\!} (Pi function) – the gamma function when offset to coincide with the factorial Rectangular function π ( n ) {\displaystyle \pi (n)\
Pi_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
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
Equation whose unknown is a function
equations have highly irregular solutions. For example, the gamma function is a function that satisfies the functional equation f ( x + 1 ) = x f ( x
Functional_equation
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
Special function defined by an integral
π ) {\displaystyle -(\Gamma (0,-\ln 2)+i\,\pi )} where Γ ( a , x ) {\displaystyle \Gamma (a,x)} is the incomplete gamma function. It must be understood
Logarithmic_integral_function
Characterization of the gamma function
In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers z {\displaystyle z} for which R e z > 0 {\displaystyle
Wielandt_theorem
Tool in multivariate statistical analysis
)},} where Γ {\displaystyle \Gamma } is the gamma function, K ν {\displaystyle K_{\nu }} is the modified Bessel function of the second kind, and ρ and
Matérn_covariance_function
Mathematical theorem on convolved binomial coefficients
b;c;1)=Γ(c)Γ(c−a−b)Γ(c−a)Γ(c−b) where 2F1 is the hypergeometric function and Γ(n+1)=n! is the gamma function. One regains the Chu–Vandermonde identity by taking a = −n
Vandermonde's_identity
Complex-differentiable (mathematical) function
holomorphic function along a loop vanishes: ∮ γ f ( z ) d z = 0. {\displaystyle \oint _{\gamma }f(z)\,\mathrm {d} z=0.} Here γ {\displaystyle \gamma } is
Holomorphic_function
Sequence of numbers ((2n) choose (n))
{\displaystyle \Gamma (x)} is the gamma function and B ( x , y ) {\displaystyle \mathrm {B} (x,y)} is the beta function. The powers of two that divide the
Central_binomial_coefficient
Probability distribution and special case of gamma distribution
incomplete gamma function and P ( s , t ) {\textstyle P(s,t)} is the regularized gamma function. In a special case of k = 2 {\displaystyle k=2} this function has
Chi-squared_distribution
Discrete probability distribution
logarithm of the Gamma function can be obtained using the lgamma function in the C standard library (C99 version) or R, the gammaln function in MATLAB or
Poisson_distribution
Two identities for Gauss sums
zeta function of a Fermat hypersurface over a finite field, which motivated the Weil conjectures. Gauss sums are analogues of the gamma function over
Hasse–Davenport_relation
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
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
Generalization of the binomial theorem to other polynomials
Kummer's theorem. By Stirling's approximation, or equivalently the log-gamma function's asymptotic expansion, log ( k n n , n , ⋯ , n ) = k n log ( k )
Multinomial_theorem
Mathematical formula involving a given set of operations
functions such as the error function or gamma function to be basic. It is possible to solve the quintic equation if general hypergeometric functions are
Closed-form_expression
Expresses a Gauss sum using a product of values of the p-adic gamma function
product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport
Gross–Koblitz_formula
Number theory expression
{n}{p^{i}}}\right\rfloor ,} where ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } is the floor function. While the sum on the right side is an infinite sum, for any particular
Legendre's_formula
High dynamic range standard that was jointly developed by the BBC and NHK
Hybrid log–gamma (HLG) is a transfer function jointly developed by the BBC and NHK for high dynamic range (HDR) display. It is backward compatible with
Hybrid_log–gamma
{z^{k}}{k^{s}}}~{\frac {\Gamma (s,kb)}{\Gamma (s)}}} where Γ(s) is the gamma function and Γ(s,x) is the upper incomplete gamma function. Since Γ(s,0)=Γ(s),
Incomplete_polylogarithm
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
Simpler variant of the Riemann zeta function
{\displaystyle \zeta (s)} denotes the Riemann zeta function and Γ ( s ) {\displaystyle \Gamma (s)} is the gamma function. The functional equation (or reflection
Riemann_xi_function
Branch of discrete mathematics
combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic
Combinatorics
Constants of the mathematical zeta function
/4)}}-{\frac {\Gamma '(1/2)}{\Gamma (1/2)}}=\log(2\pi )+{\frac {\pi }{2}}+2\log 2+\gamma \,.} The following sums can be derived from the generating function: ∑ k
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
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GAMMA FUNCTION
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GAMMA FUNCTION
GAMMA FUNCTION
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GAMMA FUNCTION
GAMMA FUNCTION
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