Definition:Gaussian Integral/Two Variables

Definition
The Gaussian Integral (of two variables) is the following definite integral, considered as a real-valued function:


 * $\phi: \left\{ \left({a,b}\right)\in \R^2: a \le b \right \} \to \R$:


 * $\phi\left({a,b}\right) = \displaystyle \int_a^b \frac 1 {\sqrt{2\pi}} \exp \left({-\frac 1 2t^2 }\right) \, \mathrm dt$

where $\exp$ is the real exponential function.

A common abuse of notation is to denote the improper integrals as:


 * $\displaystyle \phi\left({a, +\infty}\right) = \lim_{b \to +\infty} \phi\left({a,b}\right)$


 * $\displaystyle \phi\left({-\infty, b} \right) = \lim_{a \to -\infty} \phi\left({a,b}\right)$