List of integrals of Gaussian functions

In the expressions in this article,

is the standard normal probability density function,

is the corresponding cumulative distribution function (where erf is the error function), and

{\displaystyle T(h,a)=\varphi (h)\int _{0}^{a}{\frac {\varphi (hx)}{1+x^{2}}}\,dx}

is Owen's T function.

Owen[1] has an extensive list of Gaussian-type integrals; only a subset is given below.

In the previous two integrals, n!!

is the double factorial: for even n it is equal to the product of all even numbers from 2 to n, and for odd n it is the product of all odd numbers from 1 to n; additionally it is assumed that 0!!