List of integrals of Gaussian functions

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 <a href="/facts/Standard_normal/UapjjPyQ">standard normal</a> 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 <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> (where erf is the <a href="/facts/Error_function/Ca2H7pGr">error function</a>), 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 <a href="/facts/Owen%2527s_T_function/VBV5fFHW">Owen's T function</a>.
Owen has an extensive list of Gaussian-type integrals; only a subset is given below.

List of integrals of Gaussian functions open-in-new

List of integrals of Gaussian functions