DLMF:14.17.E13 (Q4894)

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DLMF:14.17.E13
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    Statements

    Defining formula
    ∫ - 1 1 ( 𝖰 ν ⁡ ( x ) ) 2 ⁢ d x = π 2 - 2 ⁢ ( 1 + cos 2 ⁡ ( ν ⁢ π ) ) ⁢ ψ ′ ⁡ ( ν + 1 ) 2 ⁢ ( 2 ⁢ ν + 1 ) , superscript subscript 1 1 superscript shorthand-Ferrers-Legendre-Q-first-kind 𝜈 𝑥 2 𝑥 superscript 𝜋 2 2 1 2 𝜈 𝜋 diffop digamma 1 𝜈 1 2 2 𝜈 1 {\displaystyle{\displaystyle\int_{-1}^{1}\left(\mathsf{Q}_{\nu}\left(x\right)% \right)^{2}\mathrm{d}x=\frac{\pi^{2}-2\left(1+{\cos^{2}}\left(\nu\pi\right)% \right)\psi'\left(\nu+1\right)}{2(2\nu+1)},}}
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    DLMF id
    DLMF:14.17.E13
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    constraint
    - 1 , - 2 , - 3 , … 1 2 3 … {\displaystyle{\displaystyle-1,-2,-3,\dots}}
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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.m2acdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aldec
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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.m2acdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aldec
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    Q11596
    Defining formula
    𝖰 ν ⁡ ( x ) = 𝖰 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-Q-first-kind 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{Q}^{0}_{\nu}\left(x\right)}}
    xml-id
    C14.S2.SS2.p2.m4aadec
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    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1ldec
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    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1hdec
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