Question

How do you find ( -3/2)factorial?

Answer 1

`(-3/2)! = Gamma(-1/2) = -2sqrt(pi)`

Explanation:

Strictly speaking, factorial is only defined for non-negative integers, but its definition is extended to other values using the `Gamma` function. For positive numbers and Complex numbers with positive Real part, we have:

`Gamma(t) = int_(x=0)^oo x^(t-1) e^(-x) dx`

Which satisfies:

`n! = Gamma(n+1)color(white)(X)` for any non-negative integer `n`.

`Gamma(1/2) = sqrt(pi)/2`

In the case of `-3/2`, we can (sort of) write:

`(-3/2)! = Gamma(-3/2+1)`

`= (Gamma(-1/2))/(-1/2)`

`= (Gamma(1/2))/(1/2*(-1/2))`

`= (sqrt(pi)/2)/(-1/4)`

`= -4(sqrt(pi)/2)`

`= -2sqrt(pi)`