Factorial Calculation for Non-Integers?

In general, $~n!~=~\displaystyle\int_0^\infty\exp\Big(-\sqrt[n]x\Big)~dx,~$ which for $~n=\dfrac12~$ yields $~\Big(\tfrac12\Big)!~=~\displaystyle\int_0^\infty e^{-x^2}~dx.~$

But the value of the Gaussian integral is known to be $\sqrt\pi~,~$ implying that $~\Big(\tfrac12\Big)!~=~\dfrac{\sqrt\pi}2,~$

since the integrand is even. Now all that's left to do is to repeatedly employ the well-known

factorial property $(n+1)!=(n+1)~n!~$ for $~n+1=4+\dfrac12,~$ and the result follows.

There is a function called the Gamma function. It is similar to the factorial as the factorial could be thought of as a special case of the gamma function.

$\Gamma(n) = (n-1)!$

or rather, when you shift it by one, as shown in the above equation.

The gamma function happens to be

$\Gamma(t) = \int_0^\infty x^{t-1} e^{-x} dx$

Calculators often use the gamma function to calculate factorials of non-natural values.

The generalization is useful when you need to extend the definition of the factorial beyond the natural numbers. For example, some probability distributions use the factorial, and the gamma function can be used to generalize them.

The factorial and gamma function both have some interesting properties in common.

For example, the factorial function can be defined recursively.

$0!=1$

$(n+1)! = (n+1) \times n!$

The gamma function also has this property

$\Gamma (1) = 1$

$\Gamma(x+1) = (x+1) \times \Gamma(x) $