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DLMF:14.15.E11 - MaRDI portal

DLMF:14.15.E11 (Q5554)

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DLMF:14.15.E11
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    Statements

    DLMF defining formula
    𝖯 ν - μ ⁡ ( cos ⁡ θ ) = 1 ν μ ⁢ ( θ sin ⁡ θ ) 1 / 2 ⁢ ( J μ ⁡ ( ( ν + 1 2 ) ⁢ θ ) + O ⁡ ( 1 ν ) ⁢ env ⁢ J μ ⁡ ( ( ν + 1 2 ) ⁢ θ ) ) , Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝜃 1 superscript 𝜈 𝜇 superscript 𝜃 𝜃 1 2 Bessel-J 𝜇 𝜈 1 2 𝜃 Big-O 1 𝜈 envelope-Bessel-J 𝜇 𝜈 1 2 𝜃 {\displaystyle{\displaystyle\mathsf{P}^{-\mu}_{\nu}\left(\cos\theta\right)=% \frac{1}{\nu^{\mu}}\left(\frac{\theta}{\sin\theta}\right)^{1/2}\left(J_{\mu}% \left(\left(\nu+\tfrac{1}{2}\right)\theta\right)+O\left(\frac{1}{\nu}\right)% \mathrm{env}\mskip-2.0mu J_{\mu}\left(\left(\nu+\tfrac{1}{2}\right)\theta% \right)\right),}}
    0 references
    Symbols used
    Bessel function of the first kind
    DLMF defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2adec
    0 references
    order not exceeding
    DLMF defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2afdec
    0 references
    Ferrers function of the first kind
    DLMF defining formula
    𝖯 ν μ ⁡ ( x ) Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E1.m2abdec
    0 references
    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
    0 references
    Q11064
    DLMF defining formula
    env ⁢ J ν ⁡ ( x ) envelope-Bessel-J 𝜈 𝑥 {\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu J_{\NVar{\nu}}\left(\NVar% {x}\right)}}
    xml-id
    C2.S8.SS4.p5.m2adec
    0 references
    sine function
    DLMF defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
    0 references
    general order
    DLMF defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1gdec
    0 references
    general degree
    DLMF defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1gdec
    0 references
    angle
    DLMF defining formula
    θ 𝜃 {\displaystyle{\displaystyle\theta}}
    xml-id
    C14.S15.XMD7.m1dec
    0 references

    Identifiers

     
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