Define Gamma function
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The gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. If n is a positive integer, The gamma function is defined for all complex numbers except the non-positive integers.
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The gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral:
{\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx}
This integral function is extended by analytic continuation to all complex numbers except the non-positive integers (where the function has simple poles), yielding the meromorphic function we call the gamma function. It has no zeroes, so the reciprocal gamma function1/Γ(z) is a holomorphic function. In fact the gamma function corresponds to the Mellin transform of the negative exponential function:
{\displaystyle \Gamma (z)=\{{\mathcal {M}}e^{-x}\}(z)}
The gamma function is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.
{\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}e^{-x}\,dx}
This integral function is extended by analytic continuation to all complex numbers except the non-positive integers (where the function has simple poles), yielding the meromorphic function we call the gamma function. It has no zeroes, so the reciprocal gamma function1/Γ(z) is a holomorphic function. In fact the gamma function corresponds to the Mellin transform of the negative exponential function:
{\displaystyle \Gamma (z)=\{{\mathcal {M}}e^{-x}\}(z)}
The gamma function is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.
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