Fractional Factorial!

I tried my best to upload an easy and elegant solution

For Help about Gamma function click here

Its easy first of all

$\Rightarrow n!=n\times (n-1)!$

$\Rightarrow \Gamma n=(n-1)!$

$\Rightarrow n!=n\Gamma n \\ \Rightarrow \Gamma(\dfrac{5}{2})=\Gamma(1+\dfrac{3}{2})$

$\Rightarrow \therefore$ as we know $\Gamma (1+n)=n\Gamma (n)$

$\Rightarrow \Gamma(1+\dfrac{3}{2})=\dfrac{3}{2} \Gamma(\dfrac{3}{2})$

Similarly $\Rightarrow \Gamma(\dfrac{3}{2})=\Gamma(1+\dfrac{1}{2})$

$\Rightarrow \therefore \Gamma(\dfrac{3}{2})=\dfrac{1}{2}\Gamma(\dfrac{1}{2})$

and we know $\Rightarrow \Gamma(\dfrac{1}{2})=\sqrt{\pi}$

so $\Rightarrow \Gamma(\dfrac{3}{2})=\dfrac{1}{2}\sqrt{\pi}=0.8864$

$\Rightarrow \therefore \Gamma(\dfrac{5}{2})=\dfrac{3}{2} \Gamma(\dfrac{3}{2}) \\ =1.329$

$\Rightarrow \therefore \dfrac{5}{2}!=\dfrac{5}{2}\cdot 1.329$

the answer is $\Rightarrow \huge\color{royalblue}{\boxed{3.323}}$

$n!=\int\limits_0^∞ x^ne^{-x} dx$

$2.5!=\int\limits_0^∞ x^{2.5}e^{-x} dx$

Applied calculator. I apologize for wanting convenience. Gamma (3.5) = 15 Sqrt (Pi)/ 8 = 3.323

2.5!=Γ(3.5) 2.5!=2.5⋅Γ(2.5) 2.5!=2.5⋅1.5⋅Γ(1.5) 2.5!=2.5⋅1.5⋅.5⋅Γ(.5) 2.5!=2.5⋅1.5⋅.5⋅ √π 2.5!=3.323