DLMF:14.17.E2 (Q4883)

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DLMF:14.17.E2
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    Statements

    Defining formula
    ∫ ( 1 - x 2 ) μ / 2 ⁢ 𝖯 ν μ ⁡ ( x ) ⁢ d x = ( 1 - x 2 ) ( μ + 1 ) / 2 ( ν - μ ) ⁢ ( ν + μ + 1 ) ⁢ 𝖯 ν μ + 1 ⁡ ( x ) , superscript 1 superscript 𝑥 2 𝜇 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝑥 superscript 1 superscript 𝑥 2 𝜇 1 2 𝜈 𝜇 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 1 𝜈 𝑥 {\displaystyle{\displaystyle\int\left(1-x^{2}\right)^{\mu/2}\mathsf{P}^{\mu}_{% \nu}\left(x\right)\mathrm{d}x=\frac{\left(1-x^{2}\right)^{(\mu+1)/2}}{(\nu-\mu% )(\nu+\mu+1)}\mathsf{P}^{\mu+1}_{\nu}\left(x\right),}}
    0 references
    DLMF id
    DLMF:14.17.E2
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    constraint
    - ν - 1 𝜈 1 {\displaystyle{\displaystyle-\nu-1}}
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    Symbols used
    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.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
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    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1adec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1adec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1adec
    0 references
     
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