DLMF:14.17.E11 (Q4892)

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DLMF:14.17.E11
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    Statements

    Defining formula
    ∫ - 1 1 ( 𝖯 ν ⁡ ( x ) ) 2 ⁢ d x = π 2 - 2 ⁢ sin 2 ⁡ ( ν ⁢ π ) ⁢ ψ ′ ⁡ ( ν + 1 ) π 2 ⁢ ( ν + 1 2 ) , superscript subscript 1 1 superscript shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 2 𝑥 superscript 𝜋 2 2 2 𝜈 𝜋 diffop digamma 1 𝜈 1 superscript 𝜋 2 𝜈 1 2 {\displaystyle{\displaystyle\int_{-1}^{1}\left(\mathsf{P}_{\nu}\left(x\right)% \right)^{2}\mathrm{d}x=\frac{\pi^{2}-2{\sin^{2}}\left(\nu\pi\right)\psi'\left(% \nu+1\right)}{\pi^{2}\left(\nu+\frac{1}{2}\right)},}}
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    DLMF id
    DLMF:14.17.E11
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    constraint
    ν ≠ - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\nu\neq-\frac{1}{2}}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
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    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
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    Q11595
    Defining formula
    𝖯 ν ⁡ ( x ) = 𝖯 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 Ferrers-Legendre-P-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{P}^{0}_{\nu}\left(x\right)}}
    xml-id
    C14.S2.SS2.p2.m2aadec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
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    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1jdec
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    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1fdec
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