DLMF:14.3.E7 (Q4696)

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DLMF:14.3.E7
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    Statements

    Defining formula
    Q ν μ ⁡ ( x ) = e μ ⁢ π ⁢ i ⁢ π 1 / 2 ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ ( x 2 - 1 ) μ / 2 2 ν + 1 ⁢ x ν + μ + 1 ⁢ 𝐅 ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 , 1 2 ⁢ ν + 1 2 ⁢ μ + 1 2 ; ν + 3 2 ; 1 x 2 ) , Legendre-Q-second-kind 𝜇 𝜈 𝑥 superscript 𝑒 𝜇 𝜋 𝑖 superscript 𝜋 1 2 Euler-Gamma 𝜈 𝜇 1 superscript superscript 𝑥 2 1 𝜇 2 superscript 2 𝜈 1 superscript 𝑥 𝜈 𝜇 1 scaled-hypergeometric-bold-F 1 2 𝜈 1 2 𝜇 1 1 2 𝜈 1 2 𝜇 1 2 𝜈 3 2 1 superscript 𝑥 2 {\displaystyle{\displaystyle Q^{\mu}_{\nu}\left(x\right)=e^{\mu\pi i}\frac{\pi% ^{1/2}\Gamma\left(\nu+\mu+1\right)\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}x^{% \nu+\mu+1}}\mathbf{F}\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+% \tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),}}
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    DLMF id
    DLMF:14.3.E7
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    constraint
    μ + ν ≠ - 1 , - 2 , - 3 , … 𝜇 𝜈 1 2 3 … {\displaystyle{\displaystyle\mu+\nu\neq-1,-2,-3,\dots}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
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    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.m2adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
    Defining formula
    𝐅 ⁡ ( a , b ; c ; z ) scaled-hypergeometric-bold-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z% }\right)}}
    xml-id
    C15.S2.E2.m2afdec
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    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1fdec
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    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1cdec
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    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1edec
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