The Q-function is defined by : $Q(x) =\frac{1}{\sqrt{2\pi}} \int_{x}^{\infty}\exp(-\frac{u^2}{2}) \ \mathrm{d}u \ \ (1).$
According to the wiki page there is an alternative form of the Q-function based on John W. Craig's work that is more useful is expressed as: $Q(x) =\frac{1}{\pi} \int_{0}^{\frac{\pi}{2}}\exp\left(-\frac{x^2}{2\sin^2(\theta)}\right) \ \mathrm{d}\theta \ \ (2).$
Craig's proove is based on probabilistic approach, there for I look for an analytic one.
any help will be appreciated.
Thanks.