DLMF:33.6.E5 (Q9537): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q12666 / rank
 
Normal rank
Property / Symbols used: Q12666 / qualifier
 
Defining formula:

ℓ ℓ {\displaystyle{\displaystyle\ell}}

\ell
Property / Symbols used: Q12666 / qualifier
 
xml-id: C33.S1.XMD2.m1ddec

Revision as of 14:24, 2 January 2020

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DLMF:33.6.E5
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    Statements

    Defining formula
    H ℓ ± ⁡ ( η , ρ ) = e ± i ⁢ θ ℓ ⁡ ( η , ρ ) ( 2 ⁢ ℓ + 1 ) ! ⁢ Γ ⁡ ( - ℓ ± i ⁢ η ) ⁢ ( ∑ k = 0 ∞ ( a ) k ( 2 ⁢ ℓ + 2 ) k ⁢ k ! ⁢ ( ∓ 2 ⁢ i ⁢ ρ ) a + k ⁢ ( ln ⁡ ( ∓ 2 ⁢ i ⁢ ρ ) + ψ ⁡ ( a + k ) - ψ ⁡ ( 1 + k ) - ψ ⁡ ( 2 ⁢ ℓ + 2 + k ) ) - ∑ k = 1 2 ⁢ ℓ + 1 ( 2 ⁢ ℓ + 1 ) ! ⁢ ( k - 1 ) ! ( 2 ⁢ ℓ + 1 - k ) ! ⁢ ( 1 - a ) k ⁢ ( ∓ 2 ⁢ i ⁢ ρ ) a - k ) , irregular-Coulomb-H plus-or-minus ℓ 𝜂 𝜌 superscript 𝑒 plus-or-minus imaginary-unit Coulomb-phase ℓ 𝜂 𝜌 2 ℓ 1 Euler-Gamma plus-or-minus ℓ imaginary-unit 𝜂 superscript subscript 𝑘 0 Pochhammer 𝑎 𝑘 Pochhammer 2 ℓ 2 𝑘 𝑘 superscript minus-or-plus 2 imaginary-unit 𝜌 𝑎 𝑘 minus-or-plus 2 imaginary-unit 𝜌 digamma 𝑎 𝑘 digamma 1 𝑘 digamma 2 ℓ 2 𝑘 superscript subscript 𝑘 1 2 ℓ 1 2 ℓ 1 𝑘 1 2 ℓ 1 𝑘 Pochhammer 1 𝑎 𝑘 superscript minus-or-plus 2 imaginary-unit 𝜌 𝑎 𝑘 {\displaystyle{\displaystyle{H^{\pm}_{\ell}}\left(\eta,\rho\right)=\frac{e^{% \pm\mathrm{i}{\theta_{\ell}}\left(\eta,\rho\right)}}{(2\ell+1)!\Gamma\left(-% \ell\pm\mathrm{i}\eta\right)}\left(\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k% }}}{{\left(2\ell+2\right)_{k}}k!}(\mp 2\mathrm{i}\rho)^{a+k}\left(\ln\left(\mp 2% \mathrm{i}\rho\right)+\psi\left(a+k\right)-\psi\left(1+k\right)-\psi\left(2% \ell+2+k\right)\right)-\sum_{k=1}^{2\ell+1}\frac{(2\ell+1)!(k-1)!}{(2\ell+1-k)% !{\left(1-a\right)_{k}}}(\mp 2\mathrm{i}\rho)^{a-k}\right),}}
    0 references
    DLMF id
    DLMF:33.6.E5
    0 references
    Symbols used
    phase of Coulomb functions
    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.m2adec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1adec
    0 references
    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5aadec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    irregular Coulomb radial functions
    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
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q12669
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C33.S1.XMD1.m1ddec
    0 references
    Q12666
    Defining formula
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C33.S1.XMD2.m1ddec
    0 references
     
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