DLMF:18.17.E7 (Q5748)

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DLMF:18.17.E7
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    Statements

    test
    ( P n ⁡ ( x ) ) 2 + 4 ⁢ π - 2 ⁢ ( 𝖰 n ⁡ ( x ) ) 2 = 4 ⁢ π - 2 ⁢ ∫ 1 ∞ Q n ⁡ ( x 2 + ( 1 - x 2 ) ⁢ t ) ⁢ ( t 2 - 1 ) - 1 2 ⁢ d t , superscript Legendre-spherical-polynomial 𝑛 𝑥 2 4 superscript 𝜋 2 superscript shorthand-Ferrers-Legendre-Q-first-kind 𝑛 𝑥 2 4 superscript 𝜋 2 superscript subscript 1 shorthand-Legendre-Q-second-kind 𝑛 superscript 𝑥 2 1 superscript 𝑥 2 𝑡 superscript superscript 𝑡 2 1 1 2 𝑡 {\displaystyle{\displaystyle\left(P_{n}\left(x\right)\right)^{2}+4\pi^{-2}% \left(\mathsf{Q}_{n}\left(x\right)\right)^{2}=4\pi^{-2}\*\int_{1}^{\infty}Q_{n% }\left(x^{2}+(1-x^{2})t\right)(t^{2}-1)^{-\frac{1}{2}}\mathrm{d}t,}}
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    DLMF id
    DLMF:18.17.E7
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    constraint
    - 1 < x < 1 1 𝑥 1 {\displaystyle{\displaystyle-1<x<1}}
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    Symbols used
    Q10920
    test
    P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r10.m2aadec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
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