DLMF:14.15.E18 (Q4864)

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DLMF:14.15.E18
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    Statements

    Defining formula
    𝑸 ν μ ⁡ ( x ) = 1 β ⁢ Γ ⁡ ( ν + μ + 1 ) ⁢ ( α 2 - y x 2 - 1 + α 2 ) 1 / 4 ⁢ K μ ⁡ ( ( ν + 1 2 ) ⁢ | y | 1 / 2 ) ⁢ ( 1 + O ⁡ ( 1 ν ) ) , associated-Legendre-black-Q 𝜇 𝜈 𝑥 1 𝛽 Euler-Gamma 𝜈 𝜇 1 superscript superscript 𝛼 2 𝑦 superscript 𝑥 2 1 superscript 𝛼 2 1 4 modified-Bessel-second-kind 𝜇 𝜈 1 2 superscript 𝑦 1 2 1 Big-O 1 𝜈 {\displaystyle{\displaystyle\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)=\frac{1}{% \beta\Gamma\left(\nu+\mu+1\right)}\left(\frac{\alpha^{2}-y}{x^{2}-1+\alpha^{2}% }\right)^{1/4}\*K_{\mu}\left(\left(\nu+\tfrac{1}{2}\right)|y|^{1/2}\right)% \left(1+O\left(\frac{1}{\nu}\right)\right),}}
    0 references
    DLMF id
    DLMF:14.15.E18
    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.m2amdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.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.m3acdec
    0 references
    modified Bessel function of the second kind
    Defining formula
    K ν ⁡ ( z ) modified-Bessel-second-kind 𝜈 𝑧 {\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E3.m2acdec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1kdec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1ndec
    0 references
    Q11590
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C14.S1.XMD8.m1ndec
    0 references
    Q11617
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C14.S15.XMD10.m1cdec
    0 references
    Q11618
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C14.S15.XMD11.m1cdec
    0 references
     
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