DLMF:14.17.E17 (Q4898)

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DLMF:14.17.E17
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    Statements

    Defining formula
    ∫ 0 π 𝖰 l ⁡ ( cos ⁡ θ ) ⁢ 𝖯 m ⁡ ( cos ⁡ θ ) ⁢ 𝖯 n ⁡ ( cos ⁡ θ ) ⁢ sin ⁡ θ ⁢ d θ = 0 , superscript subscript 0 𝜋 shorthand-Ferrers-Legendre-Q-first-kind 𝑙 𝜃 shorthand-Ferrers-Legendre-P-first-kind 𝑚 𝜃 shorthand-Ferrers-Legendre-P-first-kind 𝑛 𝜃 𝜃 𝜃 0 {\displaystyle{\displaystyle\int_{0}^{\pi}\mathsf{Q}_{l}\left(\cos\theta\right% )\mathsf{P}_{m}\left(\cos\theta\right)\mathsf{P}_{n}\left(\cos\theta\right)% \sin\theta\mathrm{d}\theta=0,}}
    0 references
    DLMF id
    DLMF:14.17.E17
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    constraint
    | m - n | < l < m + n 𝑚 𝑛 𝑙 𝑚 𝑛 {\displaystyle{\displaystyle|m-n|<l<m+n}}
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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.m2afdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1apdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3apdec
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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.m2addec
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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.m4addec
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