DLMF:33.9.E3 (Q9553)

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DLMF:33.9.E3
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    Statements

    Defining formula
    F ℓ ⁡ ( η , ρ ) = C ℓ ⁡ ( η ) ⁢ ( 2 ⁢ ℓ + 1 ) ! ( 2 ⁢ η ) 2 ⁢ ℓ + 1 ⁢ ρ - ℓ ⁢ ∑ k = 2 ⁢ ℓ + 1 ∞ b k ⁢ t k / 2 ⁢ I k ⁡ ( 2 ⁢ t ) , regular-Coulomb-F ℓ 𝜂 𝜌 Coulomb-C ℓ 𝜂 2 ℓ 1 superscript 2 𝜂 2 ℓ 1 superscript 𝜌 ℓ superscript subscript 𝑘 2 ℓ 1 subscript 𝑏 𝑘 superscript 𝑡 𝑘 2 modified-Bessel-first-kind 𝑘 2 𝑡 {\displaystyle{\displaystyle F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta% \right)\frac{(2\ell+1)!}{(2\eta)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{% \infty}b_{k}t^{k/2}I_{k}\left(\textstyle 2\sqrt{t}\right),}}
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    DLMF id
    DLMF:33.9.E3
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    constraint
    η > 0 𝜂 0 {\displaystyle{\displaystyle\eta>0}}
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    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.m2adec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
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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.m2adec
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    regular Coulomb radial function
    Defining formula
    F ℓ ⁡ ( η , ρ ) regular-Coulomb-F ℓ 𝜂 𝜌 {\displaystyle{\displaystyle F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}% \right)}}
    xml-id
    C33.S2.E3.m2aadec
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    Q12669
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C33.S1.XMD1.m1bdec
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    Q12666
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C33.S1.XMD2.m1bdec
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    Q12667
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C33.S1.XMD5.m1adec
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    Q12668
    Defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C33.S1.XMD7.m1bdec
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    Q12681
    Defining formula
    t 𝑡 {\displaystyle{\displaystyle t}}
    xml-id
    C33.S9.XMD2.m1dec
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    Q12682
    Defining formula
    b k subscript 𝑏 𝑘 {\displaystyle{\displaystyle b_{k}}}
    xml-id
    C33.S9.XMD3.m1dec
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