DLMF:14.3.E10 (Q4699)

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DLMF:14.3.E10
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    Statements

    Defining formula
    𝑸 ν μ ⁡ ( x ) = e - μ ⁢ π ⁢ i ⁢ Q ν μ ⁡ ( x ) Γ ⁡ ( ν + μ + 1 ) . associated-Legendre-black-Q 𝜇 𝜈 𝑥 superscript 𝑒 𝜇 𝜋 𝑖 Legendre-Q-second-kind 𝜇 𝜈 𝑥 Euler-Gamma 𝜈 𝜇 1 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=e^{-\mu% \pi i}\frac{Q^{\mu}_{\nu}\left(x\right)}{\Gamma\left(\nu+\mu+1\right)}.}}
    0 references
    DLMF id
    DLMF:14.3.E10
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2afdec
    0 references
    Q11470
    Defining formula
    Q ν μ ⁡ ( z ) Legendre-Q-second-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m2aadec
    0 references
    Q11592
    Defining formula
    𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1idec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1edec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1hdec
    0 references
     
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