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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. Other fractional arguments can be approximated through efficient infinite products, infinite series ...
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.
Although the factorial function is defined only for integer arguments, it can be extended to fractional arguments using the gamma function. The gamma function for half-integers is an important part of the formula for the volume of an n-dimensional ball of radius , [7] = / (+) .
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 cumulative distribution function is the regularized gamma function: (; ... – the Gamma distribution is a member of the family of Modified half-normal ...
In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers for which > by = +,as the only function defined on the half-plane := {: >} such that:
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:
The half-normal distribution is a special case of the generalized gamma distribution with d = 1, p = 2, a = . If Y has a half-normal distribution, Y-2 has a Lévy distribution; The Rayleigh distribution is a moment-tilted and scaled generalization of the half-normal distribution.