Fraction Factorial

Question

How do we find factorial of fractions? For eg: $\frac{1!}{2!}=(\frac{\pi}{4})^{\frac{1}{2}}$ Factorials are used in combinatorics and they can only be functioned on integers to give integers.Then how do we get this?

Answer 1

Related to factorials is the gamma $(\Gamma)$ function. For positve integers it satisfies: $\Gamma(n)=(n-1)!$. For non-integers it is given by: $\Gamma(t)=\int_0^\infty x^{t-1}e^{-x}dx$.

Further reading: Wikipedia