DLMF:33.2.E7 (Q9502)

From DRMF
Revision as of 14:21, 2 January 2020 by Admin (talk | contribs) (‎Changed an Item: Add constraint)
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:33.2.E7
No description defined

    Statements

    Defining formula
    H ℓ ± ⁡ ( η , ρ ) = ( ∓ i ) ℓ ⁢ e ( π ⁢ η / 2 ) ± i ⁢ σ ℓ ⁡ ( η ) ⁢ W ∓ i ⁢ η , ℓ + 1 2 ⁡ ( ∓ 2 ⁢ i ⁢ ρ ) , irregular-Coulomb-H plus-or-minus ℓ 𝜂 𝜌 superscript minus-or-plus imaginary-unit ℓ superscript 𝑒 plus-or-minus 𝜋 𝜂 2 imaginary-unit Coulomb-phase-shift ℓ 𝜂 Whittaker-confluent-hypergeometric-W minus-or-plus imaginary-unit 𝜂 ℓ 1 2 minus-or-plus 2 imaginary-unit 𝜌 {\displaystyle{\displaystyle{H^{\pm}_{\ell}}\left(\eta,\rho\right)=(\mp\mathrm% {i})^{\ell}e^{(\pi\eta/2)\pm\mathrm{i}{\sigma_{\ell}}\left(\eta\right)}W_{\mp% \mathrm{i}\eta,\ell+\frac{1}{2}}\left(\mp 2\mathrm{i}\rho\right),}}
    0 references
    DLMF id
    DLMF:33.2.E7
    0 references
    Symbols used
    Coulomb phase shift
    Defining formula
    σ ℓ ⁡ ( η ) Coulomb-phase-shift ℓ 𝜂 {\displaystyle{\displaystyle{\sigma_{\NVar{\ell}}}\left(\NVar{\eta}\right)}}
    xml-id
    C33.S2.E10.m2adec
    0 references
    Whittaker confluent hypergeometric function
    Defining formula
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    Q12666
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C33.S1.XMD2.m1fdec
    0 references
    Q12667
    Defining formula
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C33.S1.XMD5.m1cdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q9502&oldid=46134"