DLMF:14.17.E16 (Q4897): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q11589 / rank
 
Normal rank
Property / Symbols used: Q11589 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11589 / qualifier
 
xml-id: C14.S1.XMD1.m1odec

Revision as of 13:25, 2 January 2020

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DLMF:14.17.E16
No description defined

    Statements

    Defining formula
    ∫ - 1 1 𝖯 l m ⁡ ( x ) ⁢ 𝖰 n m ⁡ ( x ) ⁢ d x = ( 1 - ( - 1 ) l + n ) ⁢ ( l + m ) ! ( l - n ) ⁢ ( l + n + 1 ) ⁢ ( l - m ) ! , superscript subscript 1 1 Ferrers-Legendre-P-first-kind 𝑚 𝑙 𝑥 Ferrers-Legendre-Q-first-kind 𝑚 𝑛 𝑥 𝑥 1 superscript 1 𝑙 𝑛 𝑙 𝑚 𝑙 𝑛 𝑙 𝑛 1 𝑙 𝑚 {\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}^{m}_{l}\left(x\right)% \mathsf{Q}^{m}_{n}\left(x\right)\mathrm{d}x=\frac{\left(1-(-1)^{l+n}\right)(l+% m)!}{(l-n)(l+n+1)(l-m)!},}}
    0 references
    DLMF id
    DLMF:14.17.E16
    0 references
    constraint
    l ≠ n 𝑙 𝑛 {\displaystyle{\displaystyle l\neq n}}
    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.m2aidec
    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.m2abdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aodec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aodec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1odec
    0 references
     
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