DLMF:33.9.E6 (Q9556)

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DLMF:33.9.E6
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    Statements

    Defining formula
    G ℓ ⁡ ( η , ρ ) ∼ ρ - ℓ ( ℓ + 1 2 ) ⁢ λ ℓ ⁢ ( η ) ⁢ C ℓ ⁡ ( η ) ⁢ ∑ k = 2 ⁢ ℓ + 1 ∞ ( - 1 ) k ⁢ b k ⁢ t k / 2 ⁢ K k ⁡ ( 2 ⁢ t ) , asymptotically-equals irregular-Coulomb-G ℓ 𝜂 𝜌 superscript 𝜌 ℓ ℓ 1 2 subscript 𝜆 ℓ 𝜂 Coulomb-C ℓ 𝜂 superscript subscript 𝑘 2 ℓ 1 superscript 1 𝑘 subscript 𝑏 𝑘 superscript 𝑡 𝑘 2 modified-Bessel-second-kind 𝑘 2 𝑡 {\displaystyle{\displaystyle G_{\ell}\left(\eta,\rho\right)\sim\frac{\rho^{-% \ell}}{(\ell+\frac{1}{2})\lambda_{\ell}(\eta)C_{\ell}\left(\eta\right)}\*\sum_% {k=2\ell+1}^{\infty}(-1)^{k}b_{k}t^{k/2}K_{k}\left(\textstyle 2\sqrt{t}\right)% ,}}
    0 references
    DLMF id
    DLMF:33.9.E6
    0 references
    Symbols used
    normalizing constant for Coulomb radial functions
    Defining formula
    C ℓ ⁡ ( η ) Coulomb-C ℓ 𝜂 {\displaystyle{\displaystyle C_{\NVar{\ell}}\left(\NVar{\eta}\right)}}
    xml-id
    C33.S2.E5.m2abdec
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    asymptotic equality
    Defining formula
    ∼ asymptotically-equals {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.E1.m2adec
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    Q12670
    Defining formula
    G ℓ ⁡ ( η , ρ ) irregular-Coulomb-G ℓ 𝜂 𝜌 {\displaystyle{\displaystyle G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}% \right)}}
    xml-id
    C33.S2.E11.m1adec
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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.m2adec
    0 references
    Q12669
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C33.S1.XMD1.m1edec
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    Q12666
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C33.S1.XMD2.m1edec
    0 references
    Q12667
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C33.S1.XMD5.m1cdec
    0 references
    Q12668
    Defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C33.S1.XMD7.m1edec
    0 references
    Q12681
    Defining formula
    t 𝑡 {\displaystyle{\displaystyle t}}
    xml-id
    C33.S9.XMD2.m1bdec
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    Q12682
    Defining formula
    b k subscript 𝑏 𝑘 {\displaystyle{\displaystyle b_{k}}}
    xml-id
    C33.S9.XMD3.m1cdec
    0 references
    coefficient (locally)
    Defining formula
    λ ℓ ⁢ ( η ) subscript 𝜆 ℓ 𝜂 {\displaystyle{\displaystyle\lambda_{\ell}(\eta)}}
    xml-id
    C33.S9.XMD4.m1dec
    0 references
     
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