I'm reading this proof of $\Gamma(\frac12)=\sqrt{\pi}$, but I just can't understand this part.
$$\left[{\Gamma(\frac12)}\right]^2=4\int_0^\infty\int_0^\infty{e^{-(u^2+v^2)}}dvdu=4\int_0^{\frac{\pi}{2}}\int_0^\infty{e^{-r^2}rdrd{\theta}}$$
I think it is something about polar coordinates but unclear. I would really appreciate if someone could give me a clear explanation. Thank you :)