DLMF:14.23.E2 (Q4948): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Ferrers function of the first kind / rank
 
Normal rank
Property / Symbols used: Ferrers function of the first kind / qualifier
 
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)}}

\FerrersP[\NVar{\mu}]{\NVar{\nu}}@{\NVar{x}}
Property / Symbols used: Ferrers function of the first kind / qualifier
 
xml-id: C14.S3.E1.m2aadec

Revision as of 13:38, 2 January 2020

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DLMF:14.23.E2
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    Statements

    Defining formula
    𝑸 ν μ ⁡ ( x ± i ⁢ 0 ) = e ± μ ⁢ π ⁢ i / 2 Γ ⁡ ( ν + μ + 1 ) ⁢ ( 𝖰 ν μ ⁡ ( x ) ∓ 1 2 ⁢ π ⁢ i ⁢ 𝖯 ν μ ⁡ ( x ) ) . associated-Legendre-black-Q 𝜇 𝜈 plus-or-minus 𝑥 𝑖 0 superscript 𝑒 plus-or-minus 𝜇 𝜋 𝑖 2 Euler-Gamma 𝜈 𝜇 1 minus-or-plus Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 1 2 𝜋 𝑖 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\pm i0\right)=% \frac{e^{\pm\mu\pi i/2}}{\Gamma\left(\nu+\mu+1\right)}\left(\mathsf{Q}^{\mu}_{% \nu}\left(x\right)\mp\tfrac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(x\right)% \right).}}
    0 references
    DLMF id
    DLMF:14.23.E2
    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.m2aadec
    0 references
     
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