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The gamma function is an important special function in mathematics.Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
Such a function is known as a pseudogamma function, the most famous being the Hadamard function. [2] The gamma function, Γ(z) in blue, plotted along with Γ(z) + sin(πz) in green. Notice the intersection at positive integers. Both are valid extensions of the factorials to a meromorphic function on the complex plane.
Plot of the Barnes G aka double gamma function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D. In mathematics, the multiple gamma function is a generalization of the Euler gamma function and the Barnes G-function.
Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: [2] (,) = = (+) (+) = = (+ +). Given the rapid growth in absolute value of Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all ...
If X ~ Gamma(ν/2, 2) (in the shape–scale parametrization), then X is identical to χ 2 (ν), the chi-squared distribution with ν degrees of freedom. Conversely, if Q ~ χ 2 (ν) and c is a positive constant, then cQ ~ Gamma(ν/2, 2c). If θ = 1/α, one obtains the Schulz-Zimm distribution, which is most prominently used to model polymer ...
In q-analog theory, the -gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by Jackson (1905) .
Plot of the Barnes G aka double gamma function G(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D The Barnes G function along part of the real axis. In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers.
The function is derived by Anderson [2] from first principles who also cites earlier work by Wishart, Mahalanobis and others. There also exists a version of the multivariate gamma function which instead of a single complex number takes a -dimensional vector of complex numbers as its argument. It generalizes the above defined multivariate gamma ...