DLMF:14.9.E9 (Q4800)

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DLMF:14.9.E9
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    Statements

    Defining formula
    2 Γ ⁡ ( ν + μ + 1 ) ⁢ Γ ⁡ ( μ - ν ) ⁢ 𝖰 ν μ ⁡ ( x ) = - cos ⁡ ( ν ⁢ π ) ⁢ 𝖯 ν - μ ⁡ ( x ) + cos ⁡ ( μ ⁢ π ) ⁢ 𝖯 ν - μ ⁡ ( - x ) , 2 Euler-Gamma 𝜈 𝜇 1 Euler-Gamma 𝜇 𝜈 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜈 𝜋 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜇 𝜋 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\frac{2}{\Gamma\left(\nu+\mu+1\right)\Gamma\left(% \mu-\nu\right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)=-\cos\left(\nu\pi\right)% \mathsf{P}^{-\mu}_{\nu}\left(x\right)+\cos\left(\mu\pi\right)\mathsf{P}^{-\mu}% _{\nu}\left(-x\right),}}
    0 references
    DLMF id
    DLMF:14.9.E9
    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.m2aedec
    0 references
    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.m2agdec
    0 references
    Ferrers function of the second kind
    Defining formula
    𝖰 ν μ ⁡ ( x ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E2.m2aedec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1hdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1fdec
    0 references
     
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