DLMF:14.17.E5 (Q4886)

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DLMF:14.17.E5
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    Statements

    Defining formula
    ∫ 0 1 x σ ⁢ ( 1 - x 2 ) μ / 2 ⁢ 𝖯 ν - μ ⁡ ( x ) ⁢ d x = Γ ⁡ ( 1 2 ⁢ σ + 1 2 ) ⁢ Γ ⁡ ( 1 2 ⁢ σ + 1 ) 2 μ + 1 ⁢ Γ ⁡ ( 1 2 ⁢ σ - 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ) ⁢ Γ ⁡ ( 1 2 ⁢ σ + 1 2 ⁢ ν + 1 2 ⁢ μ + 3 2 ) , superscript subscript 0 1 superscript 𝑥 𝜎 superscript 1 superscript 𝑥 2 𝜇 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝑥 Euler-Gamma 1 2 𝜎 1 2 Euler-Gamma 1 2 𝜎 1 superscript 2 𝜇 1 Euler-Gamma 1 2 𝜎 1 2 𝜈 1 2 𝜇 1 Euler-Gamma 1 2 𝜎 1 2 𝜈 1 2 𝜇 3 2 {\displaystyle{\displaystyle\int_{0}^{1}x^{\sigma}\left(1-x^{2}\right)^{\mu/2}% \mathsf{P}^{-\mu}_{\nu}\left(x\right)\mathrm{d}x=\frac{\Gamma\left(\frac{1}{2}% \sigma+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}\sigma+1\right)}{2^{\mu+1}% \Gamma\left(\frac{1}{2}\sigma-\frac{1}{2}\nu+\frac{1}{2}\mu+1\right)\Gamma% \left(\frac{1}{2}\sigma+\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{2}\right)},}}
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    DLMF id
    DLMF:14.17.E5
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    constraint
    ℜ ⁡ σ > - 1 𝜎 1 {\displaystyle{\displaystyle\Re\sigma>-1}}
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    ℜ ⁡ μ > - 1 𝜇 1 {\displaystyle{\displaystyle\Re\mu>-1}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2addec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
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    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1ddec
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    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1ddec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1ddec
    0 references
     
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