\begin{definition}[Definition:Gaussian Integral/Two Variables]
The '''Gaussian Integral''' (of two variables) is the following definite integral, considered as a real-valued function:
:$\phi_2: \set {\tuple {a, b} \in \R^2: a \le b} \to \R$: 
:$\map {\phi_2} {a, b} = \ds \int_a^b \frac 1 {\sqrt {2 \pi} } \map \exp {-\frac {t^2} 2} \rd t$
where $\exp$ is the real exponential function.
A common abuse of notation is to denote the improper integrals as:
:$\ds \map {\phi_2} {a, +\infty} = \lim_{b \mathop \to +\infty} \map {\phi_2} {a, b}$
:$\ds \map {\phi_2} {-\infty, b} = \lim_{a \mathop \to -\infty} \map {\phi_2} {a, b}$
\end{definition}