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)!.

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