Chapter 11.6, Problem 6P

(a) By making the change of variable $t=x^2$ in the integral that defines the gamma function, show that $\Gamma \left(1/2\right)=2{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-x^2}dx}$.

(b) Use your result from (a) to show that ${\left[\Gamma \left(1/2\right)\right]}^2=4{\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-\left(x^2+y^2\right)}dx}}dy$

(c) By changing to polar coordinates, evaluate the double integral in (b0 and hence show that $\Gamma \left(1/2\right)=\sqrt{\pi }$.