DLMF:14.17.E19 (Q4900)

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DLMF:14.17.E19
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    Statements

    Defining formula
    ∫ 1 ∞ Q ν ⁡ ( x ) ⁢ Q λ ⁡ ( x ) ⁢ d x = ψ ⁡ ( λ + 1 ) - ψ ⁡ ( ν + 1 ) ( λ - ν ) ⁢ ( λ + ν + 1 ) , superscript subscript 1 shorthand-Legendre-Q-second-kind 𝜈 𝑥 shorthand-Legendre-Q-second-kind 𝜆 𝑥 𝑥 digamma 𝜆 1 digamma 𝜈 1 𝜆 𝜈 𝜆 𝜈 1 {\displaystyle{\displaystyle\int_{1}^{\infty}Q_{\nu}\left(x\right)Q_{\lambda}% \left(x\right)\mathrm{d}x=\frac{\psi\left(\lambda+1\right)-\psi\left(\nu+1% \right)}{(\lambda-\nu)(\lambda+\nu+1)},}}
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    DLMF id
    DLMF:14.17.E19
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    constraint
    ℜ ⁡ ( λ + ν ) > - 1 𝜆 𝜈 1 {\displaystyle{\displaystyle\Re(\lambda+\nu)>-1}}
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    λ 𝜆 {\displaystyle{\displaystyle\lambda}}
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    ν ≠ - 1 , - 2 , - 3 , … 𝜈 1 2 3 … {\displaystyle{\displaystyle\nu\neq-1,-2,-3,\dots}}
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    Symbols used
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ardec
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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.m2afdec
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