DLMF:14.17.E10 (Q4891): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: psi (or digamma) function / rank
 
Normal rank
Property / Symbols used: psi (or digamma) function / qualifier
 
test:

ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}

\digamma@{\NVar{z}}
Property / Symbols used: psi (or digamma) function / qualifier
 
xml-id: C5.S2.E2.m2adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aidec
Property / Symbols used
 
Property / Symbols used: Q11595 / rank
 
Normal rank
Property / Symbols used: Q11595 / qualifier
 
test:

𝖯 ν ⁡ ( x ) = 𝖯 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 Ferrers-Legendre-P-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{P}^{0}_{\nu}\left(x\right)}}

\FerrersP[]{\NVar{\nu}}@{\NVar{x}}=\FerrersP[0]{\nu}@{x}
Property / Symbols used: Q11595 / qualifier
 
xml-id: C14.S2.SS2.p2.m2adec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Q11589 / rank
 
Normal rank
Property / Symbols used: Q11589 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11589 / qualifier
 
xml-id: C14.S1.XMD1.m1idec
Property / Symbols used
 
Property / Symbols used: Q11590 / rank
 
Normal rank
Property / Symbols used: Q11590 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11590 / qualifier
 
xml-id: C14.S1.XMD8.m1edec

Latest revision as of 12:24, 2 January 2020

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Language Label Description Also known as
English
DLMF:14.17.E10
No description defined

    Statements

    test
    ∫ - 1 1 𝖯 ν ⁡ ( x ) ⁢ 𝖯 λ ⁡ ( x ) ⁢ d x = 2 ⁢ ( 2 ⁢ sin ⁡ ( ν ⁢ π ) ⁢ sin ⁡ ( λ ⁢ π ) ⁢ ( ψ ⁡ ( ν + 1 ) - ψ ⁡ ( λ + 1 ) ) + π ⁢ sin ⁡ ( ( λ - ν ) ⁢ π ) ) π 2 ⁢ ( λ - ν ) ⁢ ( λ + ν + 1 ) , superscript subscript 1 1 shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 shorthand-Ferrers-Legendre-P-first-kind 𝜆 𝑥 𝑥 2 2 𝜈 𝜋 𝜆 𝜋 digamma 𝜈 1 digamma 𝜆 1 𝜋 𝜆 𝜈 𝜋 superscript 𝜋 2 𝜆 𝜈 𝜆 𝜈 1 {\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}_{\nu}\left(x\right)\mathsf% {P}_{\lambda}\left(x\right)\mathrm{d}x=\frac{2\left(2\sin\left(\nu\pi\right)% \sin\left(\lambda\pi\right)\left(\psi\left(\nu+1\right)-\psi\left(\lambda+1% \right)\right)+\pi\sin\left((\lambda-\nu)\pi\right)\right)}{\pi^{2}(\lambda-% \nu)(\lambda+\nu+1)},}}
    0 references
    DLMF id
    DLMF:14.17.E10
    0 references
    constraint
    - ν - 1 𝜈 1 {\displaystyle{\displaystyle-\nu-1}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
    0 references
    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aidec
    0 references
    Q11595
    test
    𝖯 ν ⁡ ( x ) = 𝖯 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 Ferrers-Legendre-P-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{P}^{0}_{\nu}\left(x\right)}}
    xml-id
    C14.S2.SS2.p2.m2adec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    Q11589
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1idec
    0 references
    Q11590
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1edec
    0 references
     
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