So I've been recently arguing with my teacher about factorials. My teacher says that factorials can only be calculated for integers, because the definition of factorials is as follows:
the product of an integer and all the integers below it
But I've seen many different calculators where you can find the factorials of real numbers, like $5!$, $1.1!$, $-5.7!$, $\sqrt 2!$ or $(\tan 85)!$ and so on...
How do calculators do this? What formula do they use? It's obviously not $(n) \times (n-1) \times \cdots \times 1$ where $n$ is some integer. (In this case it's any real number)
The calculator is using the gamma function, that satisfies for natural numbers $n$ $$\Gamma(n) = (n-1)!$$ and it is given by $$\Gamma(t) = \int_0^\infty x^{t-1}e^{-x}\,\mathrm{d}x.$$ However, note that when you say $-5.7!$, that means $-(5.7!),$ not $(-5.7)!$.