DLMF:14.30.E8 (Q4970): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10768 / rank
 
Normal rank
Property / Symbols used: Q10768 / qualifier
 
test:

δ j , k Kronecker 𝑗 𝑘 {\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}

\Kroneckerdelta{\NVar{j}}{\NVar{k}}
Property / Symbols used: Q10768 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r4.m5adec
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
 
test:

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.m2aedec
Property / Symbols used
 
Property / Symbols used: complex conjugate / rank
 
Normal rank
Property / Symbols used: complex conjugate / qualifier
 
test:

z ¯ 𝑧 {\displaystyle{\displaystyle\overline{\NVar{z}}}}

\conj{\NVar{z}}
Property / Symbols used: complex conjugate / qualifier
 
xml-id: C1.S9.E11.m2aadec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: sine function / rank
 
Normal rank
Property / Symbols used: sine function / qualifier
 
test:

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

\sin@@{\NVar{z}}
Property / Symbols used: sine function / qualifier
 
xml-id: C4.S14.E1.m2abdec
Property / Symbols used
 
Property / Symbols used: spherical harmonic / rank
 
Normal rank
Property / Symbols used: spherical harmonic / qualifier
 
test:

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.m2afdec
Property / Symbols used
 
Property / Symbols used: Q11641 / rank
 
Normal rank
Property / Symbols used: Q11641 / qualifier
 
test:

m 𝑚 {\displaystyle{\displaystyle m}}

m
Property / Symbols used: Q11641 / qualifier
 
xml-id: C14.S30.XMD1.m1gdec
Property / Symbols used
 
Property / Symbols used: Q11642 / rank
 
Normal rank
Property / Symbols used: Q11642 / qualifier
 
test:

l 𝑙 {\displaystyle{\displaystyle l}}

l
Property / Symbols used: Q11642 / qualifier
 
xml-id: C14.S30.XMD2.m1gdec

Latest revision as of 12:44, 2 January 2020

No description defined
Language Label Description Also known as
English
DLMF:14.30.E8
No description defined

    Statements

    test
    ∫ 0 2 ⁢ π ∫ 0 π Y l 1 , m 1 ⁡ ( θ , ϕ ) ¯ ⁢ Y l 2 , m 2 ⁡ ( θ , ϕ ) ⁢ sin ⁡ θ ⁢ d θ ⁢ d ϕ = δ l 1 , l 2 ⁢ δ m 1 , m 2 . superscript subscript 0 2 𝜋 superscript subscript 0 𝜋 spherical-harmonic-Y subscript 𝑙 1 subscript 𝑚 1 𝜃 italic-ϕ spherical-harmonic-Y subscript 𝑙 2 subscript 𝑚 2 𝜃 italic-ϕ 𝜃 𝜃 italic-ϕ Kronecker subscript 𝑙 1 subscript 𝑙 2 Kronecker subscript 𝑚 1 subscript 𝑚 2 {\displaystyle{\displaystyle\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{% 1}},{m_{1}}}\left(\theta,\phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi% \right)\sin\theta\mathrm{d}\theta\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_% {1},m_{2}}.}}
    0 references
    DLMF id
    DLMF:14.30.E8
    0 references
    Symbols used
    Q10768
    test
    δ j , k Kronecker 𝑗 𝑘 {\displaystyle{\displaystyle\delta_{\NVar{j},\NVar{k}}}}
    xml-id
    introduction.Sx4.p1.t1.r4.m5adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
    0 references
    complex conjugate
    test
    z ¯ 𝑧 {\displaystyle{\displaystyle\overline{\NVar{z}}}}
    xml-id
    C1.S9.E11.m2aadec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    sine function
    test
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    spherical harmonic
    test
    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.m2afdec
    0 references
    Q11641
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S30.XMD1.m1gdec
    0 references
    Q11642
    test
    l 𝑙 {\displaystyle{\displaystyle l}}
    xml-id
    C14.S30.XMD2.m1gdec
    0 references
     
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