Factorial

Factorial of a number $ n $ is the product of the positive integers numbers (not null) less or equal to $ n $.

The usual notation to indicate a factorial is the exclamation mark positioned after the number. The factorial of $ n $ is noted $ n! $.

Factorial is calculated with a multiplication

$$ n!=\prod_{k=1}^n k = 1 \times 2 \times 3 \times \cdots \times n $$

Euler-Gamma is an extension of the factorial function over the complex numbers set.

$$ \Gamma(n+1) = \int_0^{+\infty} t^n \exp(-t) \rm{d}t $$

and the formula that links gamma to the factorial:

$$ \forall\,n \in \mathbb{N}, \; \Gamma(n+1)=n! $$

For computing the factorial equivalent of negative numbers, use the Gamma function.

Pour computing the factorial equivalent of fraction or decimal numbers, use the Gamma function.

The factorial algorithm with a loop:function fact(n) {
f = 1
if (n >= 2) {
for (i = 2 ; i < n; i++) {
f = f * i
}
}
return f
}

The recursive factorial algorithm: function fact(n) {
if (n <= 1)
return 1
else
return fact(n-1)*n
}

For large numbers, it is possible to estimate the value of $ n! $ with a good precision using the Stirling formula. $$ n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n $$

To calculate the product of numbers between $ n $ and $ n + m $, use factorials:

$$ \prod_{i=0}^m (n+i) = n(n+1)(n+2)\cdots(n+m) = \frac{(n+m)!}{(n-1)!} $$