Formula:KLS:01.02:05

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formula in The gamma and beta function

Formula:KLS:01.02:06 >>

${\displaystyle{\displaystyle{\displaystyle\int_{-\infty}^{\infty}{\mathrm{e}^{-\alpha^{2}x^{2}-2\beta x}}\,dx=\sqrt{\frac{\pi}{\alpha^{2}}}\,{\mathrm{e}^{\beta^{2}/\alpha^{2}}}\quad\alpha,\beta\in\mathbb{R}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\alpha\neq 0}}}$

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

Bibliography

Equation in Section 1.2 of KLS.

URL links

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formula in The gamma and beta function

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