Can we prove $\Gamma(1/2)$=$\sqrt{\pi}$ using elementary techniques .
I can easily prove properties of gamma functions like $\Gamma(n+1)$=n$\Gamma(n)$and value of $\Gamma(1)$ using integration by parts .
Most of the proves i saw used the integral $\int_0^\infty e^{-x^2}$ , wheres to solve this integral i substitute $x^2$=t and then get the answer as (1/2)$\Gamma(1/2)$=$\sqrt{\pi}/2$ becuase i am a high school student and know only elementary methods to solve integrals
Non conventional methods /some other beautiful methods (which can be elementary or near to elementary) to evaluate this are appreciated .
Best Answer
Recall the Beta function
$$\int_0^{\pi/2}\cos^{2x-1}(\theta)\sin ^{2y-1}(\theta)d \theta= \frac{1}{2}\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \tag{1}$$
now, let $x=\frac{1}{2}\,\, \text{and}\,\,y=\frac{1}{2}$ in $(1)$
$$ \begin{aligned} \int_0^{\pi/2}d \theta= \frac{1}{2}\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\right)}{\Gamma(1)}\\ \frac{\pi}{2}=\frac{1}{2}\Gamma^2\left(\frac{1}{2}\right)\\ \Gamma^2\left(\frac{1}{2}\right)=\pi\\ \Gamma\left(\frac{1}{2}\right)=\sqrt{\pi} \end{aligned} $$