What is the factorial of a negative number and a fraction?

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I have heard that factorials of negative numbers and fractions exist. Is there any proof to this fact and what value do they assume?

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The Gamma Function: $\Gamma (z) = \int_{0}^{\infty} t^{z-1} e^{-t} \mathrm {d}t $ is a generalisation of the factorial to non integers.

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