Definition:Gaussian Integral/Two Variables

Definition

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

$\phi_2: \set {\tuple {a, b} \in \R^2: a \le b} \to \R$:

$\map {\phi_2} {a, b} = \ds \int_a^b \frac 1 {\sqrt {2 \pi} } \map \exp {-\frac {t^2} 2} \rd t$

where $\exp$ is the real exponential function.

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

$\ds \map {\phi_2} {a, +\infty} = \lim_{b \mathop \to +\infty} \map {\phi_2} {a, b}$

$\ds \map {\phi_2} {-\infty, b} = \lim_{a \mathop \to -\infty} \map {\phi_2} {a, b}$

Sources

• 2001: Michael A. Bean: Probability: The Science of Uncertainty: $\S 6.3$
• 2011: Charles Henry Brase and Corrinne Pellillo Brase: Understandable Statistics (10th ed.): $\S 6.1$