DLMF:14.30.E9 (Q4971): Difference between revisions

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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: 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.m2adec
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.m2acdec
Property / Symbols used
 
Property / Symbols used: spherical harmonic / rank
 
Normal rank
Property / Symbols used: spherical harmonic / qualifier
 
Defining formula:

Y l , m ⁡ ( θ , ϕ ) spherical-harmonic-Y 𝑙 𝑚 𝜃 italic-ϕ {\displaystyle{\displaystyle Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},% \NVar{\phi}\right)}}

\sphharmonicY{\NVar{l}}{\NVar{m}}@{\NVar{\theta}}{\NVar{\phi}}
Property / Symbols used: spherical harmonic / qualifier
 
xml-id: C14.S30.E1.m2agdec
Property / Symbols used
 
Property / Symbols used: Q11641 / rank
 
Normal rank
Property / Symbols used: Q11641 / qualifier
 
Defining formula:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11641 / qualifier
 
xml-id: C14.S30.XMD1.m1hdec
Property / Symbols used
 
Property / Symbols used: Q11642 / rank
 
Normal rank
Property / Symbols used: Q11642 / qualifier
 
Defining formula:

l 𝑙 {\displaystyle{\displaystyle l}}

l
Property / Symbols used: Q11642 / qualifier
 
xml-id: C14.S30.XMD2.m1hdec

Latest revision as of 13:44, 2 January 2020

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DLMF:14.30.E9
No description defined

    Statements

    Defining formula
    𝖯 l ⁡ ( cos ⁡ θ 1 ⁢ cos ⁡ θ 2 + sin ⁡ θ 1 ⁢ sin ⁡ θ 2 ⁢ cos ⁡ ( ϕ 1 - ϕ 2 ) ) = 4 ⁢ π 2 ⁢ l + 1 ⁢ ∑ m = - l l Y l , m ⁡ ( θ 1 , ϕ 1 ) ¯ ⁢ Y l , m ⁡ ( θ 2 , ϕ 2 ) . shorthand-Ferrers-Legendre-P-first-kind 𝑙 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 1 subscript 𝜃 2 subscript italic-ϕ 1 subscript italic-ϕ 2 4 𝜋 2 𝑙 1 superscript subscript 𝑚 𝑙 𝑙 spherical-harmonic-Y 𝑙 𝑚 subscript 𝜃 1 subscript italic-ϕ 1 spherical-harmonic-Y 𝑙 𝑚 subscript 𝜃 2 subscript italic-ϕ 2 {\displaystyle{\displaystyle\mathsf{P}_{l}\left(\cos\theta_{1}\cos\theta_{2}+% \sin\theta_{1}\sin\theta_{2}\cos\left(\phi_{1}-\phi_{2}\right)\right)=\frac{4% \pi}{2l+1}\sum_{m=-l}^{l}\overline{Y_{{l},{m}}\left(\theta_{1},\phi_{1}\right)% }Y_{{l},{m}}\left(\theta_{2},\phi_{2}\right).}}
    0 references
    DLMF id
    DLMF:14.30.E9
    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
    complex conjugate
    Defining formula
    z ¯ 𝑧 {\displaystyle{\displaystyle\overline{\NVar{z}}}}
    xml-id
    C1.S9.E11.m2abdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    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.m2adec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2acdec
    0 references
    spherical harmonic
    Defining formula
    Y l , m ⁡ ( θ , ϕ ) spherical-harmonic-Y 𝑙 𝑚 𝜃 italic-ϕ {\displaystyle{\displaystyle Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},% \NVar{\phi}\right)}}
    xml-id
    C14.S30.E1.m2agdec
    0 references
    Q11641
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S30.XMD1.m1hdec
    0 references
    Q11642
    Defining formula
    l 𝑙 {\displaystyle{\displaystyle l}}
    xml-id
    C14.S30.XMD2.m1hdec
    0 references
     
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