How demonstrate the Craig representation for the Gaussian probability function? Well Done @Didier, About the Craig's proof I refered to this [pdf document][1].
When I searching I found a book called : "Probability Distributions Involving Gaussian Random Variables" state that :
<<The form in (l) is not readily obtainable by a change of variables directly in (2). However, by first extending (1) to two dimensions (x and y) where one of the dimensions (y) is integrated over the half plane, a change of variables from rectangular to polar coordinates readily produces (2).>> pg.123.
If this is clear for you, how can be done?
[1]:wsl.stanford.edu/~ee359/craig.pdf