DLMF:14.17.E17 (Q4898): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2afdec
Property / Symbols used
 
Property / Symbols used: cosine function / rank
 
Normal rank
Property / Symbols used: cosine function / qualifier
 
Defining formula:

cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}

\cos@@{\NVar{z}}
Property / Symbols used: cosine function / qualifier
 
xml-id: C4.S14.E2.m2acdec
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.m1apdec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3apdec
Property / Symbols used
 
Property / Symbols used: Q11595 / rank
 
Normal rank
Property / Symbols used: Q11595 / qualifier
 
Defining formula:

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.m2addec
Property / Symbols used
 
Property / Symbols used: Q11596 / rank
 
Normal rank
Property / Symbols used: Q11596 / qualifier
 
Defining formula:

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

\FerrersQ[]{\NVar{\nu}}@{\NVar{x}}=\FerrersQ[0]{\nu}@{x}
Property / Symbols used: Q11596 / qualifier
 
xml-id: C14.S2.SS2.p2.m4addec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
Defining formula:

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

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2aedec
Property / Symbols used
 
Property / Symbols used: Q11593 / rank
 
Normal rank
Property / Symbols used: Q11593 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11593 / qualifier
 
xml-id: C14.S1.XMD5.m1edec
Property / Symbols used
 
Property / Symbols used: Q11602 / rank
 
Normal rank
Property / Symbols used: Q11602 / qualifier
 
Defining formula:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11602 / qualifier
 
xml-id: C14.S1.XMD6.m1edec

Latest revision as of 13:26, 2 January 2020

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DLMF:14.17.E17
No description defined

    Statements

    Defining formula
    ∫ 0 π 𝖰 l ⁡ ( cos ⁡ θ ) ⁢ 𝖯 m ⁡ ( cos ⁡ θ ) ⁢ 𝖯 n ⁡ ( cos ⁡ θ ) ⁢ sin ⁡ θ ⁢ d θ = 0 , superscript subscript 0 𝜋 shorthand-Ferrers-Legendre-Q-first-kind 𝑙 𝜃 shorthand-Ferrers-Legendre-P-first-kind 𝑚 𝜃 shorthand-Ferrers-Legendre-P-first-kind 𝑛 𝜃 𝜃 𝜃 0 {\displaystyle{\displaystyle\int_{0}^{\pi}\mathsf{Q}_{l}\left(\cos\theta\right% )\mathsf{P}_{m}\left(\cos\theta\right)\mathsf{P}_{n}\left(\cos\theta\right)% \sin\theta\mathrm{d}\theta=0,}}
    0 references
    DLMF id
    DLMF:14.17.E17
    0 references
    constraint
    | m - n | < l < m + n 𝑚 𝑛 𝑙 𝑚 𝑛 {\displaystyle{\displaystyle|m-n|<l<m+n}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2afdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1apdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3apdec
    0 references
    Q11595
    Defining formula
    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.m2addec
    0 references
    Q11596
    Defining formula
    𝖰 ν ⁡ ( x ) = 𝖰 ν 0 ⁡ ( x ) shorthand-Ferrers-Legendre-Q-first-kind 𝜈 𝑥 Ferrers-Legendre-Q-first-kind 0 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=% \mathsf{Q}^{0}_{\nu}\left(x\right)}}
    xml-id
    C14.S2.SS2.p2.m4addec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aedec
    0 references
    Q11593
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1edec
    0 references
    Q11602
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1edec
    0 references
     
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