Value of Gaussian Integral over Reals

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Theorem

Let $\phi_2$ be the Gaussian Integral of Two Variables:

$\phi_2: \left\{ \left({a,b}\right)\in \R^2: a \le b \right \} \to \R$:
$\phi_2\left({a,b}\right) = \displaystyle \int_a^b \frac 1 {\sqrt{2\pi}} \exp \left({-\frac 1 2t^2 }\right) \, \mathrm dt$


Then the value of the improper integral of $\phi_2$ over the reals is one:

$\phi_2\left({-\infty,+\infty}\right) = \displaystyle \int_{\to -\infty}^{\to +\infty} \frac 1 {\sqrt{2\pi}} \exp \left({-\frac 1 2t^2 }\right) \, \mathrm dt = 1$

Equivalently:

$\displaystyle \lim_{x \to +\infty} \phi_1 \left({x}\right) = 1$

where $\phi_1$ is the Gaussian Integral of One Variable.


First Part

$\displaystyle \int_{\R} e^{-x^2} \, \mathrm dx = \sqrt{\pi}$


Second Part

Value of Gaussian Integral over Reals/Second Part