DLMF:14.2.E7 (Q4685)

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DLMF:14.2.E7
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    Statements

    Defining formula
    𝒲 ⁡ { P ν - μ ⁡ ( x ) , P ν μ ⁡ ( x ) } = 𝒲 ⁡ { 𝖯 ν - μ ⁡ ( x ) , 𝖯 ν μ ⁡ ( x ) } = 2 ⁢ sin ⁡ ( μ ⁢ π ) π ⁢ ( 1 - x 2 ) , Wronskian Legendre-P-first-kind 𝜇 𝜈 𝑥 Legendre-P-first-kind 𝜇 𝜈 𝑥 Wronskian Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 2 𝜇 𝜋 𝜋 1 superscript 𝑥 2 {\displaystyle{\displaystyle\mathscr{W}\left\{P^{-\mu}_{\nu}\left(x\right),P^{% \mu}_{\nu}\left(x\right)\right\}=\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}% \left(x\right),\mathsf{P}^{\mu}_{\nu}\left(x\right)\right\}=\frac{2\sin\left(% \mu\pi\right)}{\pi\left(1-x^{2}\right)},}}
    0 references
    DLMF id
    DLMF:14.2.E7
    0 references
    Symbols used
    Ferrers function of the first kind
    Defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2addec
    0 references
    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
    0 references
    Q10865
    Defining formula
    𝒲 Wronskian {\displaystyle{\displaystyle\mathscr{W}}}
    xml-id
    C1.S13.Px2.p1.m3acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1fdec
    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.m1fdec
    0 references
     
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