Formula:6327

Digital Library of Mathematical Functions ID 18.10.E6

${\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(x^{2}\right)=\frac{2(-1)^{n}}{\pi^{\frac{1}{2}}\Gamma\left(\alpha+\tfrac{1}{2}\right)n!}\*\int_{0}^{\infty}\int_{0}^{\pi}{(x^{2}-r^{2}+2ixr\cos\phi)^{n}}\*e^{-r^{2}}r^{2\alpha+1}(\sin\phi)^{2\alpha}\mathrm{d}\phi\mathrm{d}r,}}$

Constraint(s)

${\displaystyle{\displaystyle\alpha>-\frac{1}{2}}}$

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
• ${\displaystyle{\displaystyle\pi}}$: 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
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

Resources that cite this formula

No records found.