DLMF:14.9.E8 (Q4799)

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DLMF:14.9.E8
No description defined

    Statements

    Defining formula
    1 2 ⁢ π ⁢ sin ⁡ ( ( ν - μ ) ⁢ π ) ⁢ 𝖯 ν - μ ⁡ ( x ) = - cos ⁡ ( ( ν - μ ) ⁢ π ) ⁢ 𝖰 ν - μ ⁡ ( x ) - 𝖰 ν - μ ⁡ ( - x ) , 1 2 𝜋 𝜈 𝜇 𝜋 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 𝜈 𝜇 𝜋 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\tfrac{1}{2}\pi\sin\left((\nu-\mu)\pi\right)% \mathsf{P}^{-\mu}_{\nu}\left(x\right)=-\cos\left((\nu-\mu)\pi\right)\mathsf{Q}% ^{-\mu}_{\nu}\left(x\right)-\mathsf{Q}^{-\mu}_{\nu}\left(-x\right),}}
    0 references
    DLMF id
    DLMF:14.9.E8
    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.m2afdec
    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.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2addec
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    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1gdec
    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.m1gdec
    0 references
     
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