Formula:6463

Digital Library of Mathematical Functions ID 18.17.E25

${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{2}x^{2}}\mathit{He}_{n}\left(x\right)\mathit{He}_{n+2m}\left(x\right)\cos\left(xy\right)\mathrm{d}x=(-1)^{m}\sqrt{\tfrac{1}{2}\pi}n!\,y^{2m}e^{-\frac{1}{2}y^{2}}L^{(2m)}_{n}\left(y^{2}\right),}}$

Symbols List

• ${\displaystyle{\displaystyle\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)}}$: Hermite polynomial
• ${\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Laguerre (or generalized Laguerre) polynomial
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle y}}$: real variable
• ${\displaystyle{\displaystyle m}}$: nonnegative integer
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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