DLMF:14.9.E1 (Q4791)

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DLMF:14.9.E1
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    Statements

    Defining formula
    π ⁢ sin ⁡ ( μ ⁢ π ) 2 ⁢ Γ ⁡ ( ν - μ + 1 ) ⁢ 𝖯 ν - μ ⁡ ( x ) = - 1 Γ ⁡ ( ν + μ + 1 ) ⁢ 𝖰 ν μ ⁡ ( x ) + cos ⁡ ( μ ⁢ π ) Γ ⁡ ( ν - μ + 1 ) ⁢ 𝖰 ν - μ ⁡ ( x ) . 𝜋 𝜇 𝜋 2 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 1 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜇 𝜋 Euler-Gamma 𝜈 𝜇 1 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\frac{\pi\sin\left(\mu\pi\right)}{2\Gamma\left(\nu% -\mu+1\right)}\mathsf{P}^{-\mu}_{\nu}\left(x\right)=-\frac{1}{\Gamma\left(\nu+% \mu+1\right)}\mathsf{Q}^{\mu}_{\nu}\left(x\right)+\frac{\cos\left(\mu\pi\right% )}{\Gamma\left(\nu-\mu+1\right)}\mathsf{Q}^{-\mu}_{\nu}\left(x\right).}}
    0 references
    DLMF id
    DLMF:14.9.E1
    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.m2adec
    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.m2adec
    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.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
     
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