DLMF:14.18.E3 (Q4904): Difference between revisions

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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.m2abdec

Revision as of 13:27, 2 January 2020

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DLMF:14.18.E3
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    Statements

    Defining formula
    𝖰 ν ⁡ ( cos ⁡ θ 1 ⁢ cos ⁡ θ 2 + sin ⁡ θ 1 ⁢ sin ⁡ θ 2 ⁢ cos ⁡ ϕ ) = 𝖯 ν ⁡ ( cos ⁡ θ 1 ) ⁢ 𝖰 ν ⁡ ( cos ⁡ θ 2 ) + 2 ⁢ ∑ m = 1 ∞ ( - 1 ) m ⁢ 𝖯 ν - m ⁡ ( cos ⁡ θ 1 ) ⁢ 𝖰 ν m ⁡ ( cos ⁡ θ 2 ) ⁢ cos ⁡ ( m ⁢ ϕ ) . shorthand-Ferrers-Legendre-Q-first-kind 𝜈 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 1 subscript 𝜃 2 italic-ϕ shorthand-Ferrers-Legendre-P-first-kind 𝜈 subscript 𝜃 1 shorthand-Ferrers-Legendre-Q-first-kind 𝜈 subscript 𝜃 2 2 superscript subscript 𝑚 1 superscript 1 𝑚 Ferrers-Legendre-P-first-kind 𝑚 𝜈 subscript 𝜃 1 Ferrers-Legendre-Q-first-kind 𝑚 𝜈 subscript 𝜃 2 𝑚 italic-ϕ {\displaystyle{\displaystyle\mathsf{Q}_{\nu}\left(\cos\theta_{1}\cos\theta_{2}% +\sin\theta_{1}\sin\theta_{2}\cos\phi\right)=\mathsf{P}_{\nu}\left(\cos\theta_% {1}\right)\mathsf{Q}_{\nu}\left(\cos\theta_{2}\right)+2\sum_{m=1}^{\infty}(-1)% ^{m}\mathsf{P}^{-m}_{\nu}\left(\cos\theta_{1}\right)\mathsf{Q}^{m}_{\nu}\left(% \cos\theta_{2}\right)\cos\left(m\phi\right).}}
    0 references
    DLMF id
    DLMF:14.18.E3
    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.m2abdec
    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.m2adec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    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.m2abdec
    0 references
     
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