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

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

• You cannot prove that the factorials of negative numbers exist, since you do not have a definition for them. You should be asking «are there sensible definitions for factorials of negative numbers and fractions?» and if he answer is affirmative —and it is— then there is nothing to be proved: you do not prove definitions. – Mariano Suárez-Álvarez Feb 13 '17 at 8:02

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.