Gaussian Integral
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Theorem
- $\ds \int_{-\infty}^\infty e^{-x^2} \rd x = \sqrt \pi$
Proof
| \(\ds \int_{-\infty}^\infty e^{-x^2} \rd x\) | \(=\) | \(\ds 2 \int_0^\infty e^{-x^2} \rd x\) | Definite Integral of Even Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 \sqrt \pi} 2\) | Integral to Infinity of Exponential of -t^2 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt \pi\) |
$\blacksquare$