Derivation of Craig´s formula for Gaussian probability function

A recent work provides two proofs for Craig´s formula, which is an integral formula for the Gaussian probability density function. It also highlights the fact that the original proof of the Craig formula is somewhat unclear. The authors of the work provide two proofs, one based on the Stieltjes transform of a Gaussian pulse and another based on the moment generation function (MGF) of a unit Gaussian random variable (GRV). Another integral formula for Q(x) based on the characteristic function (CHF) of a unit GRV is also provided. This formula can be applied to the analysis of coherent predetection equal-gain combining diversity receivers