Question

How do I find the factorial of a negative number?

Answer 1

It depends.

Explanation:

Factorial as such is only defined for non-negative integers with the familiar recursive definition:

`0! = 1`
`(n+1)! = (n+1)n!` for `n >= 0`

There are a couple of extensions of the definition of factorial to cover a larger domain.

Euler's gamma function

The most mainstream extension of the definition of factorial is given by Euler's gamma function,

For positive integers:

`Gamma(n) = (n-1)!`

For any complex number `t` with a positive real part:

`Gamma(t) = int_0^oo x^(t-1)e^(-x) dx`

Then we can use the identity `Gamma(t+1) = t Gamma(t)` to extend the definition to all values except negative integers (which would entail division by `0`).

Roman factorial

This extends the definition of factorial to the negative integers as follows:

`|__n~|!` `= n!` for `n >= 0`

`|__n~|!` `= (-1)^(-n-1)/((-n-1)!)` for `n < 0`