DLMF:14.8.E7 (Q4781)

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DLMF:14.8.E7
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    Statements

    Defining formula
    P ν μ ⁡ ( x ) ∼ 1 Γ ⁡ ( 1 - μ ) ⁢ ( 2 x - 1 ) μ / 2 , asymptotically-equals Legendre-P-first-kind 𝜇 𝜈 𝑥 1 Euler-Gamma 1 𝜇 superscript 2 𝑥 1 𝜇 2 {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)\sim\frac{1}{\Gamma% \left(1-\mu\right)}\left(\frac{2}{x-1}\right)^{\mu/2},}}
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    DLMF id
    DLMF:14.8.E7
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    constraint
    μ ≠ 1 , 2 , 3 , … 𝜇 1 2 3 … {\displaystyle{\displaystyle\mu\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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    Q11481
    Defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1adec
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    asymptotic equality
    Defining formula
    ∼ asymptotically-equals {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.E1.m2aedec
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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.m1ddec
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    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1fdec
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