0.5! ......

The gamma function serves as an extension of the factorial function beyond the non-negative integer. And one of the functional equations is $\Gamma (s) = (s-1)!$ . Therefore,

$\begin{aligned} \left(\frac 12\right) ! & = \Gamma \left(\frac 32\right) & \small \color{#3D99F6} \text{Using the identity: }\Gamma (s+1) = s\Gamma (s) \\ & = \frac 12 \Gamma \left(\frac 12\right) & \small \color{#3D99F6} \text{Note that }\Gamma \left(\frac 12\right) = \sqrt \pi \\ & = \frac {\sqrt \pi}2 \\ & = \boxed{0.8862269...} \end{aligned}$

Factorials are actually only defined for integers more than or equal to $0$ so technically it should be "not possible",but , the gamma function which is the "extension" to factorials yields $\displaystyle \frac{\sqrt{\pi}}{2}$ . A little bit misleading though!

we know, Γ(n+1) = n Γ(n)

if n = 1/2

Γ(3/2) = Γ(1/2)/2

=> Γ(3/2) = √π/2

=> (1/2)! = √π/2

cliché..

$\begin{aligned} & \left(\dfrac{1}{2}\right)! \\ = & \Pi\left(\dfrac{1}{2}\right) \\ = & \displaystyle \int_{0}^{\infty} t^{1/2}e^{-t} dt \\ = & \dfrac{\sqrt{\pi}}{2} \\ \approx & \boxed{0.8862269...} \\ \end{aligned}$