DLMF:14.20.E18 (Q4939)

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DLMF:14.20.E18
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    Statements

    Defining formula
    𝖰 ^ - 1 2 + i ⁢ τ - μ ⁡ ( x ) = σ ⁢ ( μ , τ ) ⁢ ( α 2 + η 1 + α 2 - x 2 ) 1 / 4 ⁢ K μ ⁡ ( τ ⁢ η 1 / 2 ) ⁢ ( 1 + O ⁡ ( 1 / τ ) ) , Ferrers-conical-legendre-Q-hat 𝜇 1 2 𝑖 𝜏 𝑥 𝜎 𝜇 𝜏 superscript superscript 𝛼 2 𝜂 1 superscript 𝛼 2 superscript 𝑥 2 1 4 modified-Bessel-second-kind 𝜇 𝜏 superscript 𝜂 1 2 1 Big-O 1 𝜏 {\displaystyle{\displaystyle\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+i\tau}% \left(x\right)=\sigma(\mu,\tau)\left(\frac{\alpha^{2}+\eta}{1+\alpha^{2}-x^{2}% }\right)^{1/4}K_{\mu}\left(\tau\eta^{1/2}\right)\*\left(1+O\left(\ifrac{1}{% \tau}\right)\right),}}
    0 references
    DLMF id
    DLMF:14.20.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.m2acdec
    0 references
    conical function
    Defining formula
    𝖰 ^ - 1 2 + i ⁢ τ - μ ⁡ ( x ) Ferrers-conical-legendre-Q-hat 𝜇 1 2 𝑖 𝜏 𝑥 {\displaystyle{\displaystyle\widehat{\mathsf{Q}}^{-\NVar{\mu}}_{\NVar{-\frac{1% }{2}+i\tau}}\left(\NVar{x}\right)}}
    xml-id
    C14.S20.E2.m2ajdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2axdec
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    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.m2aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1udec
    0 references
    Q11630
    Defining formula
    τ 𝜏 {\displaystyle{\displaystyle\tau}}
    xml-id
    C14.S1.XMD3.m1ydec
    0 references
    Q11591
    Defining formula
    μ 𝜇 {\displaystyle{\displaystyle\mu}}
    xml-id
    C14.S1.XMD7.m1mdec
    0 references
    Q11633
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S20.XMD5.m1adec
    0 references
    Q11634
    Defining formula
    σ ⁢ ( μ , τ ) 𝜎 𝜇 𝜏 {\displaystyle{\displaystyle\sigma(\mu,\tau)}}
    xml-id
    C14.S20.XMD6.m1adec
    0 references
    Q11635
    Defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C14.S20.XMD7.m1adec
    0 references
     
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