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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.
In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. [1] The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. [2]
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.
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 ...
The probability density function of Gamma distribution for different parameters. See below for the spec and Gnuplot source code. Date: 26 June 2010, 18:31 (UTC) Source: Gamma_distribution_pdf.png; Author: Gamma_distribution_pdf.png: MarkSweep and Cburnett; derivative work: Autopilot (talk)
A Weibull distribution is a generalized gamma ... value sets an initial failure-free time ... "A statistical distribution function of wide applicability" (PDF ...
The reciprocal is sometimes used as a starting point for numerical computation of the gamma function, and a few software libraries provide it separately from the regular gamma function. Karl Weierstrass called the reciprocal gamma function the "factorielle" and used it in his development of the Weierstrass factorization theorem.
The quantile function can be found by noting that (;,,) = ((/)) where is the cumulative distribution function of the gamma distribution with parameters = / and =. The quantile function is then given by inverting F {\displaystyle F} using known relations about inverse of composite functions , yielding: