DLMF:14.15.E9 (Q4855)

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DLMF:14.15.E9
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    Statements

    Defining formula
    𝑸 ν μ ⁡ ( x ) = ( π 2 ) 1 / 2 ⁢ ( e μ ) ν + ( 1 / 2 ) ⁢ ( 1 - α 1 + α ) μ / 2 ⁢ ( 1 - α 2 ) - ( ν / 2 ) - ( 1 / 4 ) ⁢ ( α 2 + η 2 α 2 ⁢ ( x 2 - 1 ) + 1 ) 1 / 4 ⁢ I ν + 1 2 ⁡ ( μ ⁢ η ) ⁢ ( 1 + O ⁡ ( 1 μ ) ) , associated-Legendre-black-Q 𝜇 𝜈 𝑥 superscript 𝜋 2 1 2 superscript 𝑒 𝜇 𝜈 1 2 superscript 1 𝛼 1 𝛼 𝜇 2 superscript 1 superscript 𝛼 2 𝜈 2 1 4 superscript superscript 𝛼 2 superscript 𝜂 2 superscript 𝛼 2 superscript 𝑥 2 1 1 1 4 modified-Bessel-first-kind 𝜈 1 2 𝜇 𝜂 1 Big-O 1 𝜇 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\left(% \frac{\pi}{2}\right)^{1/2}\left(\frac{e}{\mu}\right)^{\nu+(1/2)}\left(\frac{1-% \alpha}{1+\alpha}\right)^{\mu/2}\*\left(1-\alpha^{2}\right)^{-(\nu/2)-(1/4)}% \left(\frac{\alpha^{2}+\eta^{2}}{\alpha^{2}\left(x^{2}-1\right)+1}\right)^{1/4% }\*I_{\nu+\frac{1}{2}}\left(\mu\eta\right)\left(1+O\left(\frac{1}{\mu}\right)% \right),}}
    0 references
    DLMF id
    DLMF:14.15.E9
    0 references
    Symbols used
    order not exceeding
    Defining formula
    O ⁡ ( x ) Big-O 𝑥 {\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
    xml-id
    C2.S1.E3.m2aedec
    0 references
    Q11592
    Defining formula
    𝑸 ν μ ⁡ ( z ) associated-Legendre-black-Q 𝜇 𝜈 𝑧 {\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(% \NVar{z}\right)}}
    xml-id
    C14.S21.SS1.p1.m3aadec
    0 references
    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
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1fdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1fdec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1fdec
    0 references
    Q11611
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S15.XMD3.m1edec
    0 references
     
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