DLMF:14.17.E4 (Q4885): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1acdec

Revision as of 13:22, 2 January 2020

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DLMF:14.17.E4
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    Statements

    Defining formula
    ∫ x ( 1 - x 2 ) 3 / 2 ⁢ 𝖯 ν μ ⁡ ( x ) ⁢ 𝖰 ν μ ⁡ ( x ) ⁢ d x = 1 ( 1 - 4 ⁢ μ 2 ) ⁢ ( 1 - x 2 ) 1 / 2 ⁢ ( ( 1 - 2 ⁢ μ 2 + 2 ⁢ ν ⁢ ( ν + 1 ) ) ⁢ 𝖯 ν μ ⁡ ( x ) ⁢ 𝖰 ν μ ⁡ ( x ) + ( 2 ⁢ ν + 1 ) ⁢ ( μ - ν - 1 ) ⁢ x ⁢ ( 𝖯 ν μ ⁡ ( x ) ⁢ 𝖰 ν + 1 μ ⁡ ( x ) + 𝖯 ν + 1 μ ⁡ ( x ) ⁢ 𝖰 ν μ ⁡ ( x ) ) + 2 ⁢ ( μ - ν - 1 ) 2 ⁢ 𝖯 ν + 1 μ ⁡ ( x ) ⁢ 𝖰 ν + 1 μ ⁡ ( x ) ) , 𝑥 superscript 1 superscript 𝑥 2 3 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝑥 1 1 4 superscript 𝜇 2 superscript 1 superscript 𝑥 2 1 2 1 2 superscript 𝜇 2 2 𝜈 𝜈 1 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 2 𝜈 1 𝜇 𝜈 1 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 1 𝑥 Ferrers-Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 2 superscript 𝜇 𝜈 1 2 Ferrers-Legendre-P-first-kind 𝜇 𝜈 1 𝑥 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 1 𝑥 {\displaystyle{\displaystyle\int\frac{x}{\left(1-x^{2}\right)^{3/2}}\mathsf{P}% ^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)\mathrm{d}x=% \frac{1}{\left(1-4\mu^{2}\right)\left(1-x^{2}\right)^{1/2}}\left((1-2\mu^{2}+2% \nu(\nu+1))\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x% \right)+(2\nu+1)(\mu-\nu-1)x(\mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{% \mu}_{\nu+1}\left(x\right)+\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{% \mu}_{\nu}\left(x\right))+2(\mu-\nu-1)^{2}\mathsf{P}^{\mu}_{\nu+1}\left(x% \right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)\right),}}
    0 references
    DLMF id
    DLMF:14.17.E4
    0 references
    constraint
    μ ≠ ± 1 2 𝜇 plus-or-minus 1 2 {\displaystyle{\displaystyle\mu\neq\pm\frac{1}{2}}}
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    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.m2acdec
    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.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
    0 references
     
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