For the gamma function of ordinals, see Veblen function. For the gamma distribution in statistics, see Gamma distribution. For the function used in video and image color representations, see Gamma correction.
In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n,
Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral:
The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.[clarification needed]
The gamma function has no zeros, so the reciprocal gamma function 1/Γ(z) is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function:
Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.
the gammafunction (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex...
In mathematics, the upper and lower incomplete gammafunctions are types of special functions which arise as solutions to various mathematical problems...
distribution functions of the gamma distribution vary based on the chosen parameterization, both offering insights into the behavior of gamma-distributed...
{\displaystyle \Gamma } is used as a symbol for: In mathematics, the gammafunction (usually written as Γ {\displaystyle \Gamma } -function) is an extension...
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gammafunction and to binomial...
mathematics, the inverse gammafunction Γ − 1 ( x ) {\displaystyle \Gamma ^{-1}(x)} is the inverse function of the gammafunction. In other words, y = Γ...
The gammafunction is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer...
In mathematics, the digamma function is defined as the logarithmic derivative of the gammafunction: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )...
Gamma correction or gamma is a nonlinear operation used to encode and decode luminance or tristimulus values in video or still image systems. Gamma correction...
reciprocal gammafunction is the function f ( z ) = 1 Γ ( z ) , {\displaystyle f(z)={\frac {1}{\Gamma (z)}},} where Γ(z) denotes the gammafunction. Since...
gammafunction Γ N {\displaystyle \Gamma _{N}} is a generalization of the Euler gammafunction and the Barnes G-function. The double gammafunction was...
gammafunction Γp is a generalization of the gammafunction. It is useful in multivariate statistics, appearing in the probability density function of...
factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gammafunction. Many other notable functions and...
{d} x} is the gammafunction. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ >...
\mathbb {C} } defined as the (m + 1)th derivative of the logarithm of the gammafunction: ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln Γ ( z )...
the Gammafunction. The Hankel contour is used to evaluate integrals such as the Gammafunction, the Riemann zeta function, and other Hankel functions (which...
\Gamma (n)=(n-1)!} . When the gammafunction is evaluated at half-integers, the result contains π. For example, Γ ( 1 / 2 ) = π {\displaystyle \Gamma (1/2)={\sqrt...
{\displaystyle \Pi (x)\,\!} (Pi function) – the gammafunction when offset to coincide with the factorial Rectangular function You might also be looking for:...
[further explanation needed] In terms of the regularized gammafunction P and the incomplete gammafunction, erf x = sgn x ⋅ P ( 1 2 , x 2 ) = sgn x π γ...
\end{aligned}}} may be derived using integration by parts. The gammafunction is an example of a special function, defined as an improper integral for z > 0 {\displaystyle...
_{m=0}^{\infty }{\frac {(-1)^{m}}{m!\,\Gamma (m+\alpha +1)}}{\left({\frac {x}{2}}\right)}^{2m+\alpha },} where Γ(z) is the gammafunction, a shifted generalization...
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...
Many values of the theta function and especially of the shown phi function can be represented in terms of the gammafunction: φ ( exp ( − 2 π ) ) =...
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