DLMF:14.8.E9 (Q4783)

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DLMF:14.8.E9
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    Statements

    test
    𝑸 Ξ½ ⁑ ( x ) = - ln ⁑ ( x - 1 ) 2 ⁒ Ξ“ ⁑ ( Ξ½ + 1 ) + 1 2 ⁒ ln ⁑ 2 - Ξ³ - ψ ⁑ ( Ξ½ + 1 ) Ξ“ ⁑ ( Ξ½ + 1 ) + O ⁑ ( x - 1 ) , shorthand-associated-Legendre-black-Q 𝜈 π‘₯ π‘₯ 1 2 Euler-Gamma 𝜈 1 1 2 2 digamma 𝜈 1 Euler-Gamma 𝜈 1 Big-O π‘₯ 1 {\displaystyle{\displaystyle\boldsymbol{Q}_{\nu}\left(x\right)=-\frac{\ln\left% (x-1\right)}{2\Gamma\left(\nu+1\right)}+\frac{\frac{1}{2}\ln 2-\gamma-\psi% \left(\nu+1\right)}{\Gamma\left(\nu+1\right)}+O\left(x-1\right),}}
    0 references
    DLMF id
    DLMF:14.8.E9
    0 references
    constraint
    Ξ½ β‰  - 1 , - 2 , - 3 , … 𝜈 1 2 3 … {\displaystyle{\displaystyle\nu\neq-1,-2,-3,\dots}}
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    Symbols used
    order not exceeding
    test
    O ⁑ ( x ) Big-O π‘₯ {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aadec
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    gamma function
    test
    Ξ“ ⁑ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
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    Euler’s constant
    test
    Ξ³ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2aadec
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    psi (or digamma) function
    test
    ψ ⁑ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2aadec
    0 references
    principal branch of logarithm function
    test
    ln ⁑ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
     
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