Question

What is the factorial of -23?

Answer 1

Undefined, unless you mean Roman factorial...

`color(black)(⌊-23⌉!) = 1/(22!) = 1/1124000727777607680000`

Explanation:

Strictly speaking, factorial is only defined for non-negative integers.

The normal, recursive definition is:

`{ (0! = 1), (n! = nxx(n-1)! color(white)(xx) " for " n >= 1) :}`

`color(white)()`
Gamma function

The normal method used to extend the definition of factorials to other numbers is using the Gamma function:

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

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

This integral converges for positive Real values of `t` and Complex values of `t` with a positive Real part.

The Gamma function can be extended to other Real and Complex numbers by analytic continuation.

The only values of `t` for which it is not possible to analytically continue `Gamma(t)` are `0, -1, -2, -3,...`, corresponding to factorials of negative integers.

So this method yields no value for `(-23)!`

`color(white)()`
Roman factorial

Another extension of the definition of factorial, which does include negative numbers is the Roman factorial:

`color(black)(⌊n⌉!) = { (n!, " for " n >= 0), ((-1)^(-n-1)/((-1-n)!), " for " n < 0) :}`

The Roman factorial is used in the definition of the harmonic logarithm.

We find:

`color(black)(⌊-23⌉!) = (-1)^(-1+23)/((-1+23)!) = 1/(22!) = 1/1124000727777607680000`