DLMF:14.2.E10 (Q4688)

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DLMF:14.2.E10
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    Statements

    Defining formula
    𝒲 ⁡ { P ν μ ⁡ ( x ) , Q ν μ ⁡ ( x ) } = - e μ ⁢ π ⁢ i ⁢ Γ ⁡ ( ν + μ + 1 ) Γ ⁡ ( ν - μ + 1 ) ⁢ ( x 2 - 1 ) , Wronskian Legendre-P-first-kind 𝜇 𝜈 𝑥 Legendre-Q-second-kind 𝜇 𝜈 𝑥 superscript 𝑒 𝜇 𝜋 𝑖 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜈 𝜇 1 superscript 𝑥 2 1 {\displaystyle{\displaystyle\mathscr{W}\left\{P^{\mu}_{\nu}\left(x\right),Q^{% \mu}_{\nu}\left(x\right)\right\}=-e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+1\right)\left(x^{2}-1\right)},}}
    0 references
    DLMF id
    DLMF:14.2.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.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.m1abdec
    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.m2adec
    0 references
    Q10865
    Defining formula
    𝒲 Wronskian {\displaystyle{\displaystyle\mathscr{W}}}
    xml-id
    C1.S13.Px2.p1.m3afdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
     
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