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DLMF:18.10.E6 - MaRDI portal

DLMF:18.10.E6 (Q6327)

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DLMF:18.10.E6
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    Statements

    DLMF defining formula
    L n ( α ) ⁡ ( x 2 ) = 2 ⁢ ( - 1 ) n π 1 2 ⁢ Γ ⁡ ( α + 1 2 ) ⁢ n ! ⁢ ∫ 0 ∞ ∫ 0 π ( x 2 - r 2 + 2 ⁢ i ⁢ x ⁢ r ⁢ cos ⁡ ϕ ) n ⁢ e - r 2 ⁢ r 2 ⁢ α + 1 ⁢ ( sin ⁡ ϕ ) 2 ⁢ α ⁢ d ϕ ⁢ d r , Laguerre-polynomial-L 𝛼 𝑛 superscript 𝑥 2 2 superscript 1 𝑛 superscript 𝜋 1 2 Euler-Gamma 𝛼 1 2 𝑛 superscript subscript 0 superscript subscript 0 𝜋 superscript superscript 𝑥 2 superscript 𝑟 2 2 𝑖 𝑥 𝑟 italic-ϕ 𝑛 superscript 𝑒 superscript 𝑟 2 superscript 𝑟 2 𝛼 1 superscript italic-ϕ 2 𝛼 italic-ϕ 𝑟 {\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,}}
    0 references
    constraint
    α > - 1 2 𝛼 1 2 {\displaystyle{\displaystyle\alpha>-\frac{1}{2}}}
    0 references
    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Laguerre (or generalized Laguerre) polynomial
    DLMF defining formula
    L n ( α ) ⁡ ( x ) Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}% \right)}}
    xml-id
    C18.S3.T1.t1.r12.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    sine function
    DLMF defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2addec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C18.S1.XMD6.m1edec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C18.S2.XMD3.m1dec
    0 references

    Identifiers

     
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