DLMF:14.8.E13 (Q4787)

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DLMF:14.8.E13
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    Statements

    Defining formula
    P ν μ ⁡ ( x ) ∼ Γ ⁡ ( - ν - 1 2 ) π 1 / 2 ⁢ Γ ⁡ ( - μ - ν ) ⁢ ( 2 ⁢ x ) ν + 1 , asymptotically-equals Legendre-P-first-kind 𝜇 𝜈 𝑥 Euler-Gamma 𝜈 1 2 superscript 𝜋 1 2 Euler-Gamma 𝜇 𝜈 superscript 2 𝑥 𝜈 1 {\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)\sim\frac{\Gamma\left(% -\nu-\frac{1}{2}\right)}{\pi^{1/2}\Gamma\left(-\mu-\nu\right)(2x)^{\nu+1}},}}
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    DLMF id
    DLMF:14.8.E13
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    constraint
    ℜ ⁡ ν < - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu<-\tfrac{1}{2}}}
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    ℜ ⁡ ν < - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu<-\tfrac{1}{2}}}
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    ν + μ ≠ 0 , 1 , 2 , … 𝜈 𝜇 0 1 2 … {\displaystyle{\displaystyle\nu+\mu\neq 0,1,2,\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.m2aidec
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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.m1acdec
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