DLMF:33.10.E2 (Q10252)

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DLMF:33.10.E2
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    Statements

    DLMF defining formula
    H ℓ ± ⁡ ( η , ρ ) ∼ exp ⁡ ( ± i ⁢ θ ℓ ⁡ ( η , ρ ) ) , asymptotically-equals irregular-Coulomb-H plus-or-minus ℓ 𝜂 𝜌 plus-or-minus imaginary-unit Coulomb-phase ℓ 𝜂 𝜌 {\displaystyle{\displaystyle{H^{\pm}_{\ell}}\left(\eta,\rho\right)\sim\exp% \left(\pm\mathrm{i}{\theta_{\ell}}\left(\eta,\rho\right)\right),}}
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    Symbols used
    phase of Coulomb functions
    DLMF defining formula
    θ ℓ ⁡ ( η , ρ ) Coulomb-phase ℓ 𝜂 𝜌 {\displaystyle{\displaystyle{\theta_{\NVar{\ell}}}\left(\NVar{\eta},\NVar{\rho% }\right)}}
    xml-id
    C33.S2.E9.m2aadec
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    asymptotic equality
    DLMF defining formula
    ∼ asymptotically-equals {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.E1.m2adec
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    exponential function
    DLMF defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
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    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    irregular Coulomb radial functions
    DLMF defining formula
    H ℓ ± ⁡ ( η , ρ ) irregular-Coulomb-H plus-or-minus ℓ 𝜂 𝜌 {\displaystyle{\displaystyle{H^{\NVar{\pm}}_{\NVar{\ell}}}\left(\NVar{\eta},% \NVar{\rho}\right)}}
    xml-id
    C33.S2.E7.m2adec
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    nonnegative integer
    DLMF defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C33.S1.XMD2.m1adec
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    nonnegative real variable
    DLMF defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C33.S1.XMD5.m1adec
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    real parameter
    DLMF defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C33.S1.XMD7.m1adec
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    Identifiers

     
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