Formula:KLS:09.12:26

<< Formula:KLS:09.12:25

formula in Laguerre

Formula:KLS:09.12:27 >>

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\Gamma\left(\alpha+1\right)}\int_{0}^{\infty}\frac{L^{\alpha}_{n}\left(y\right)}{L^{\alpha}_{n}\left(0\right)}{\mathrm{e}^{-y}}y^{\alpha}{\mathrm{e}^{\mathrm{i}xy}}dy={\mathrm{i}^{n}}\frac{y^{n}}{(\mathrm{i}y+1)^{n+\alpha+1}}}}}$

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}$ : Laguerre (or generalized Laguerre) polynomial : http://dlmf.nist.gov/18.3#T1.t1.r27
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 9.12 of KLS.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:KLS:09.12:25

formula in Laguerre

Formula:KLS:09.12:27 >>