DLMF:13.21.E6 (Q4614)

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DLMF:13.21.E6
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    Statements

    Defining formula
    M - κ , μ ⁡ ( 4 ⁢ κ ⁢ x ) = 2 ⁢ Γ ⁡ ( 2 ⁢ μ + 1 ) κ μ - 1 2 ⁢ ( x ⁢ ζ 1 + x ) 1 4 ⁢ I 2 ⁢ μ ⁡ ( 4 ⁢ κ ⁢ ζ 1 2 ) ⁢ ( 1 + O ⁡ ( κ - 1 ) ) , Whittaker-confluent-hypergeometric-M 𝜅 𝜇 4 𝜅 𝑥 2 Euler-Gamma 2 𝜇 1 superscript 𝜅 𝜇 1 2 superscript 𝑥 𝜁 1 𝑥 1 4 modified-Bessel-first-kind 2 𝜇 4 𝜅 superscript 𝜁 1 2 1 Big-O superscript 𝜅 1 {\displaystyle{\displaystyle M_{-\kappa,\mu}\left(4\kappa x\right)=\frac{2% \Gamma\left(2\mu+1\right)}{\kappa^{\mu-\frac{1}{2}}}\left(\frac{x\zeta}{1+x}% \right)^{\frac{1}{4}}I_{2\mu}\left(4\kappa\zeta^{\frac{1}{2}}\right){\left(1+O% \left(\kappa^{-1}\right)\right)},}}
    0 references
    DLMF id
    DLMF:13.21.E6
    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.m2addec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    M κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-M 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E2.m2aadec
    0 references
    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.m2adec
    0 references
    Q11566
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1edec
    0 references
     
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