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DLMF:14.12.E8 - MaRDI portal

DLMF:14.12.E8 (Q5526)

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DLMF:14.12.E8
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    Statements

    DLMF defining formula
    P n m ⁡ ( x ) = 2 m ⁢ m ! ⁢ ( n + m ) ! ⁢ ( x 2 - 1 ) m / 2 ( 2 ⁢ m ) ! ⁢ ( n - m ) ! ⁢ π ⁢ ∫ 0 π ( x + ( x 2 - 1 ) 1 / 2 ⁢ cos ⁡ ϕ ) n - m ⁢ ( sin ⁡ ϕ ) 2 ⁢ m ⁢ d ϕ , Legendre-P-first-kind 𝑚 𝑛 𝑥 superscript 2 𝑚 𝑚 𝑛 𝑚 superscript superscript 𝑥 2 1 𝑚 2 2 𝑚 𝑛 𝑚 𝜋 superscript subscript 0 𝜋 superscript 𝑥 superscript superscript 𝑥 2 1 1 2 italic-ϕ 𝑛 𝑚 superscript italic-ϕ 2 𝑚 italic-ϕ {\displaystyle{\displaystyle P^{m}_{n}\left(x\right)=\frac{2^{m}m!(n+m)!\left(% x^{2}-1\right)^{m/2}}{(2m)!(n-m)!\pi}\int_{0}^{\pi}\left(x+\left(x^{2}-1\right% )^{1/2}\cos\phi\right)^{n-m}(\sin\phi)^{2m}\mathrm{d}\phi,}}
    0 references
    constraint
    n ≥ m 𝑛 𝑚 {\displaystyle{\displaystyle n\geq m}}
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    Symbols used
    associated Legendre function of the first kind
    DLMF defining formula
    P ν μ ⁡ ( z ) Legendre-P-first-kind 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m1acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
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    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
    0 references
    Q10729
    DLMF defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
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    sine function
    DLMF defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    real variable
    DLMF defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1edec
    0 references
    nonnegative integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C14.S1.XMD5.m1adec
    0 references
    nonnegative integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1dec
    0 references

    Identifiers

     
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