Formula:KLS:09.12:26

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1 Γ ⁡ ( α + 1 ) ⁢ ∫ 0 ∞ L n α ⁡ ( y ) L n α ⁡ ( 0 ) ⁢ e - y ⁢ y α ⁢ e i ⁢ x ⁢ y ⁢ 𝑑 y = i n ⁢ y n ( i ⁢ y + 1 ) n + α + 1 1 Euler-Gamma 𝛼 1 superscript subscript 0 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑦 generalized-Laguerre-polynomial-L 𝛼 𝑛 0 𝑦 superscript 𝑦 𝛼 imaginary-unit 𝑥 𝑦 differential-d 𝑦 imaginary-unit 𝑛 superscript 𝑦 𝑛 superscript imaginary-unit 𝑦 1 𝑛 𝛼 1 {\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

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Symbols List

Γ Γ {\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
L n ( α ) superscript subscript 𝐿 𝑛 𝛼 {\displaystyle{\displaystyle{\displaystyle L_{n}^{(\alpha)}}}}  : Laguerre (or generalized Laguerre) polynomial : http://dlmf.nist.gov/18.3#T1.t1.r27
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 9.12 of KLS.

URL links

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