Definition:Gaussian Integral/Two Variables

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


 * $\phi_2: \left\{{\left({a, b}\right) \in \R^2: a \le b}\right\} \to \R$:


 * $\phi_2 \left({a, b}\right) = \displaystyle \int_a^b \frac 1 {\sqrt{2 \pi}} \exp \left({- \frac {t^2} 2}\right) \ \mathrm d t$

where $\exp$ is the real exponential function.

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


 * $\displaystyle \phi_2 \left({a, +\infty}\right) = \lim_{b \mathop \to +\infty} \phi_2 \left({a, b}\right)$


 * $\displaystyle \phi_2 \left({-\infty, b} \right) = \lim_{a \mathop \to -\infty} \phi_2 \left({a, b}\right)$