What is the meaning and value of $(-0.5!)$?

Question

What is the meaning and value of $(-0.5!)$? I got the value as $- \frac {\sqrt \pi}{2}$or $-0.8862269$. I have no idea what this means. Much appreciate your answers.

Answer 1

Usually we extend the factorial to values other than nonnegative integers by identifying it with the Gamma function. This is superficially justified by the identity $n!=\Gamma(n+1)$ for nonnegative integers $n$ (though there is really more to the story than that). The Gamma function is defined as

$$\Gamma(z)=\int_0^\infty e^{-t} t^{z-1} dt$$

if $\mathrm{Re}(z)>0$. Otherwise it is defined by analytic continuation of one of its functional equations, such as $z\Gamma(z)=\Gamma(z+1)$. In your case you want $\Gamma(1/2)$, which is in the range where the integral makes sense, and it is famously equal to $\sqrt{\pi}$ as can be seen by changing variables to convert the integral into the Gaussian integral.

Answer 2

The usual definition of the factorial is coherent only for a domain of natural numbers. But it is convenient sometimes to have a function which is defined on a much larger domain (e.g., the real or complex numbers) and agrees with the factorial on the naturals. This function should be some sort of sensible extension of the factorial. We have such a function, the Gamma function, and that's what's being used.

You can read about the Gamma function on Wikipedia -- https://en.wikipedia.org/wiki/Gamma_function

Answer 3

For the Gamma function we have $$\Gamma (n)=(n-1)!$$

Thus $$ (-0.5!) =\Gamma (0.5) =$$

$$\int_0^\infty e^{-t} t^{-0.5} dt = 1.772453...$$

The exact value of the integral is known to be $\sqrt \pi$

If we like to get $-\frac{\sqrt\pi}2$ as an answer, we compute $-(0.5!)$