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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