Synopsis:

\[ \int_{0}^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}. \]

Properties:

• The right hand side \( \frac{\sqrt{\pi}}{2} \) is a transcendental number.
• The definite integral \( \int_{0}^\infty e^{-x^2} dx \) can be used to write the following (1).

\[ \sum_{1}^8 e^{-x^2} + (\int_{0}^{p} e^{-x^2} dx) - \pi, \, \forall p \in \mathbb{N}, \, p \neq 0. \]

Amends:

After some tinkering, I would like to document this identity: \[ \sum_{n}^{p} e^{-x^2} + (\int_{n}^{p} e^{-x^2} dx) - \pi. \] Where \( 3=n \in \mathbb{N}, 18=p \in \mathbb{N} \).