Formula:KLS:09.15:30

<< Formula:KLS:09.15:29

formula in Hermite

Orthogonal Polynomials

${\displaystyle{\displaystyle{\displaystyle\frac{{\mathrm{i}^{n}}}{2\sqrt{\pi}}\int_{-\infty}^{\infty}y^{n}{\mathrm{e}^{-\frac{1}{4}y^{2}}}{\mathrm{e}^{-\mathrm{i}xy}}dy=H_{n}\left(x\right){\mathrm{e}^{-x^{2}}}}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : Hermite polynomial ${\displaystyle{\displaystyle{\displaystyle H_{n}}}}$ : http://dlmf.nist.gov/18.3#T1.t1.r28

Bibliography

Equation in Section 9.15 of KLS.

URL links

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<< Formula:KLS:09.15:29

formula in Orthogonal Polynomials

Orthogonal Polynomials