DLMF:14.30.E3 (Q4965)

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DLMF:14.30.E3
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    Statements

    Defining formula
    Y l , m ⁡ ( θ , ϕ ) = ( - 1 ) l + m 2 l ⁢ l ! ⁢ ( ( l - m ) ! ⁢ ( 2 ⁢ l + 1 ) 4 ⁢ π ⁢ ( l + m ) ! ) 1 / 2 ⁢ e i ⁢ m ⁢ ϕ ⁢ ( sin ⁡ θ ) m ⁢ ( d d ( cos ⁡ θ ) ) l + m ⁡ ( sin ⁡ θ ) 2 ⁢ l . spherical-harmonic-Y 𝑙 𝑚 𝜃 italic-ϕ superscript 1 𝑙 𝑚 superscript 2 𝑙 𝑙 superscript 𝑙 𝑚 2 𝑙 1 4 𝜋 𝑙 𝑚 1 2 superscript 𝑒 𝑖 𝑚 italic-ϕ superscript 𝜃 𝑚 superscript derivative 𝜃 𝑙 𝑚 superscript 𝜃 2 𝑙 {\displaystyle{\displaystyle Y_{{l},{m}}\left(\theta,\phi\right)=\frac{(-1)^{l% +m}}{2^{l}l!}\left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\left% (\sin\theta\right)^{m}\*\left(\frac{\mathrm{d}}{\mathrm{d}(\cos\theta)}\right)% ^{l+m}\left(\sin\theta\right)^{2l}.}}
    0 references
    DLMF id
    DLMF:14.30.E3
    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.m2aadec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
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    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
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    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.m2aadec
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    Q11641
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S30.XMD1.m1bdec
    0 references
    Q11642
    Defining formula
    l 𝑙 {\displaystyle{\displaystyle l}}
    xml-id
    C14.S30.XMD2.m1bdec
    0 references
     
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