In the expressions in this article,
φ ( x ) = 1 2 π e − 1 2 x 2 {\displaystyle \varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}}
is the standard normal probability density function,
Φ ( x ) = ∫ − ∞ x φ ( t ) d t = 1 2 [ 1 + erf ( x 2 ) ] {\displaystyle \Phi (x)=\int _{-\infty }^{x}\varphi (t)\,dt={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right]}
is the corresponding cumulative distribution function (where erf is the error function), and
T ( h , a ) = φ ( h ) ∫ 0 a φ ( h x ) 1 + x 2 d x {\displaystyle T(h,a)=\varphi (h)\int _{0}^{a}{\frac {\varphi (hx)}{1+x^{2}}}\,dx}
is Owen's T function. Owen has an extensive list of Gaussian-type integrals; only a subset is given below.
Indefinite integrals
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!! = (−1)!! = 1.
Definite integrals
References