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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.
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. The gamma function has no zeros, so the reciprocal gamma function 1 / Γ(z) is an entire function.
The double gamma function was studied by Barnes (1901). At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied these further in Barnes (1904). Double gamma functions are closely related to the q-gamma function, and triple gamma functions are related to the elliptic gamma function.
By choosing an appropriate g (typically a small integer), only some 5–10 terms of the series are needed to compute the gamma function with typical single or double floating-point precision. If a fixed g is chosen, the coefficients can be calculated in advance and, thanks to partial fraction decomposition , the sum is recast into the following ...
In mathematics, the gamma function (usually written as -function) is an extension of the factorial to complex numbers [13] In mathematics, the upper incomplete gamma function [14] The Christoffel symbols in differential geometry [15]
The upper incomplete gamma function for some values of s: 0 (blue), 1 (red), 2 (green), 3 (orange), 4 (purple). Plot of the regularized incomplete gamma function Q(2,z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
For all positive integers, ! = (+), where Γ denotes the gamma function. However, the gamma function, unlike the factorial, is more broadly defined for all complex numbers other than non-positive integers; nevertheless, Stirling's formula may still be applied.
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) .