Subjective Type

How do I find the factorial of a negative number?

Solution

Factorial as such is only defined for non-negative integers with the familiar recursive definition:
$$ 0! = 1 $$
$$ ( n +1)! = ( n+1)n! $$ for $$ n \ge 0 $$
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:

$$ T(n) = (n - 1) ! $$

For any complex number t with a positive real part:

$$ T (t) = \int^{ \infty}_0 x^{ t -1} e^{-x} dx $$

Then we can use the identity $$ T( t +1) = tT (t) $$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 $$ n \ge 0 $$

$$ [n]! = \frac {( -1)^{ -n-1} }{( -n-1)! } $$ for $$ n < 0 $$




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