DLMF:18.10.E5 (Q5629)

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DLMF:18.10.E5
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    Statements

    Defining formula
    P n ⁡ ( cos ⁡ θ ) = 1 π ⁢ ∫ 0 π ( cos ⁡ θ + i ⁢ sin ⁡ θ ⁢ cos ⁡ ϕ ) n ⁢ d ϕ . Legendre-spherical-polynomial 𝑛 𝜃 1 𝜋 superscript subscript 0 𝜋 superscript 𝜃 𝑖 𝜃 italic-ϕ 𝑛 italic-ϕ {\displaystyle{\displaystyle P_{n}\left(\cos\theta\right)=\frac{1}{\pi}\int_{0% }^{\pi}(\cos\theta+i\sin\theta\cos\phi)^{n}\mathrm{d}\phi.}}
    0 references
    DLMF id
    DLMF:18.10.E5
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    Symbols used
    Q10920
    Defining formula
    P n ⁡ ( x ) Legendre-spherical-polynomial 𝑛 𝑥 {\displaystyle{\displaystyle P_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C18.S3.T1.t1.r10.m2aadec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
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