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The gamma function $ \Gamma(x) $is the extension of the factorial function, with its argument shifted by one, to the complex plane.

For any complex number with a positive real part, the gamma function is defined by the integral $ \Gamma(z) = \int_{0}^{\infty} x^{z-1} e^{-x} dx $. Values of the gamma function elsewhere in the complex plane are found through analytic continuation, or by using the identity $ z \Gamma (z) = \Gamma (z+1) $.

Alternate FormsEdit

The gamma function can be written in terms of the factorial function for nonzero integer values as $ \Gamma(x) = \left( x-1\right)! $.