DLMF:18.17.E32

MaRDI QIDQ6470

• Label: DLMF:18.17.E32

Digital Library of Mathematical Functions ID 18.17.E32

${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-ax}x^{\nu-1-2n}L^{(\nu-1-2n)}_{2n}\left(ax\right)\cos\left(xy\right)\mathrm{d}x=\frac{(-1)^{n}\Gamma\left(\nu\right)}{2(2n)!}y^{2n}\left((a+iy)^{-\nu}+(a-iy)^{-\nu}\right),}}$

Constraint(s)

${\displaystyle{\displaystyle a>0}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)}}$: Laguerre (or generalized Laguerre) polynomial
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle y}}$: real variable
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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