DLMF:14.30.E5 (Q4967): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10818 / rank
 
Normal rank
Property / Symbols used: Q10818 / qualifier
 
Defining formula:

ℤ {\displaystyle{\displaystyle\mathbb{Z}}}

\Integers
Property / Symbols used: Q10818 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r20.m2adec

Revision as of 13:43, 2 January 2020

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DLMF:14.30.E5
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    Statements

    Defining formula
    Y l , m ⁡ ( 1 2 ⁢ π , ϕ ) = { ( - 1 ) ( l + m ) / 2 ⁢ e i ⁢ m ⁢ ϕ 2 l ⁢ ( 1 2 ⁢ l - 1 2 ⁢ m ) ! ⁢ ( 1 2 ⁢ l + 1 2 ⁢ m ) ! ⁢ ( ( l - m ) ! ⁢ ( l + m ) ! ⁢ ( 2 ⁢ l + 1 ) 4 ⁢ π ) 1 / 2 , 1 2 ⁢ l + 1 2 ⁢ m ∈ ℤ , 0 , 1 2 ⁢ l + 1 2 ⁢ m ∉ ℤ . spherical-harmonic-Y 𝑙 𝑚 1 2 𝜋 italic-ϕ cases superscript 1 𝑙 𝑚 2 superscript 𝑒 𝑖 𝑚 italic-ϕ superscript 2 𝑙 1 2 𝑙 1 2 𝑚 1 2 𝑙 1 2 𝑚 superscript 𝑙 𝑚 𝑙 𝑚 2 𝑙 1 4 𝜋 1 2 1 2 𝑙 1 2 𝑚 0 1 2 𝑙 1 2 𝑚 {\displaystyle{\displaystyle Y_{{l},{m}}\left(\tfrac{1}{2}\pi,\phi\right)=% \begin{cases}\dfrac{(-1)^{(l+m)/2}e^{im\phi}}{2^{l}\left(\frac{1}{2}l-\frac{1}% {2}m\right)!\left(\frac{1}{2}l+\frac{1}{2}m\right)!}\left(\dfrac{(l-m)!(l+m)!(% 2l+1)}{4\pi}\right)^{1/2},&\frac{1}{2}l+\frac{1}{2}m\in\mathbb{Z},\\ 0,&\frac{1}{2}l+\frac{1}{2}m\notin\mathbb{Z}.\end{cases}}}
    0 references
    DLMF id
    DLMF:14.30.E5
    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.m2acdec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q10818
    Defining formula
    ℤ {\displaystyle{\displaystyle\mathbb{Z}}}
    xml-id
    introduction.Sx4.p2.t1.r20.m2adec
    0 references
     
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