DLMF:14.18.E2 (Q4903): Difference between revisions

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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.m1adec
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.m1dec
Property / Symbols used
 
Property / Symbols used: Q11623 / rank
 
Normal rank
Property / Symbols used: Q11623 / qualifier
 
Defining formula:

θ 1 subscript 𝜃 1 {\displaystyle{\displaystyle\theta_{1}}}

\theta_{1}
Property / Symbols used: Q11623 / qualifier
 
xml-id: C14.S18.XMD1.m1adec
Property / Symbols used
 
Property / Symbols used: Q11624 / rank
 
Normal rank
Property / Symbols used: Q11624 / qualifier
 
Defining formula:

ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}

\phi
Property / Symbols used: Q11624 / qualifier
 
xml-id: C14.S18.XMD2.m1adec

Latest revision as of 13:27, 2 January 2020

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DLMF:14.18.E2
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    Statements

    Defining formula
    𝖯 n ⁡ ( cos ⁡ θ 1 ⁢ cos ⁡ θ 2 + sin ⁡ θ 1 ⁢ sin ⁡ θ 2 ⁢ cos ⁡ ϕ ) = ∑ m = - n n ( - 1 ) m ⁢ 𝖯 n - m ⁡ ( cos ⁡ θ 1 ) ⁢ 𝖯 n m ⁡ ( cos ⁡ θ 2 ) ⁢ cos ⁡ ( m ⁢ ϕ ) . shorthand-Ferrers-Legendre-P-first-kind 𝑛 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 1 subscript 𝜃 2 italic-ϕ superscript subscript 𝑚 𝑛 𝑛 superscript 1 𝑚 Ferrers-Legendre-P-first-kind 𝑚 𝑛 subscript 𝜃 1 Ferrers-Legendre-P-first-kind 𝑚 𝑛 subscript 𝜃 2 𝑚 italic-ϕ {\displaystyle{\displaystyle\mathsf{P}_{n}\left(\cos\theta_{1}\cos\theta_{2}+% \sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\sum_{m=-n}^{n}(-1)^{m}\mathsf{P}^% {-m}_{n}\left(\cos\theta_{1}\right)\mathsf{P}^{m}_{n}\left(\cos\theta_{2}% \right)\cos\left(m\phi\right).}}
    0 references
    DLMF id
    DLMF:14.18.E2
    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.m2aadec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aadec
    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.m2aadec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    Q11593
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1adec
    0 references
    Q11602
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1dec
    0 references
    Q11623
    Defining formula
    θ 1 subscript 𝜃 1 {\displaystyle{\displaystyle\theta_{1}}}
    xml-id
    C14.S18.XMD1.m1adec
    0 references
    Q11624
    Defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C14.S18.XMD2.m1adec
    0 references
     
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