Search results
Results From The WOW.Com Content Network
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.
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 ...
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 ...
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.
where () is the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Higher-dimensional versions of this theorem also appear in quantum physics through Feynman diagrams. [2] A similar result was also obtained by Glaisher. [3]
In q-analog theory, the -gamma function, or basic gamma function, is a ... December 2-5, 1993, vol. 119, Boston: Birkhäuser Verlag, ...