Formula:KLS:01.02:04

<< Formula:KLS:01.02:03

formula in The gamma and beta function

Formula:KLS:01.02:05 >>

${\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}\expe^{-x^{2}}\,dx=2\int_{0}^{\infty}\expe^{-x^{2}}\,dx=\int_{0}^{\infty}t^{-1/2}\expe^{-t}\,dt=\Gamma\left(1/2\right)=\sqrt{\cpi}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1

Bibliography

Equation in Section 1.2 of KLS.

URL links

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<< Formula:KLS:01.02:03

formula in The gamma and beta function

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