DLMF:33.12.E6 (Q9604): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q12670 / rank
 
Normal rank
Property / Symbols used: Q12670 / qualifier
 
Defining formula:

G ℓ ⁡ ( η , ρ ) irregular-Coulomb-G ℓ 𝜂 𝜌 {\displaystyle{\displaystyle G_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}% \right)}}

\irregCoulombG{\NVar{\ell}}@{\NVar{\eta}}{\NVar{\rho}}
Property / Symbols used: Q12670 / qualifier
 
xml-id: C33.S2.E11.m1abdec
Property / Symbols used
 
Property / Symbols used: regular Coulomb radial function / rank
 
Normal rank
Property / Symbols used: regular Coulomb radial function / qualifier
 
Defining formula:

F ℓ ⁡ ( η , ρ ) regular-Coulomb-F ℓ 𝜂 𝜌 {\displaystyle{\displaystyle F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}% \right)}}

\regCoulombF{\NVar{\ell}}@{\NVar{\eta}}{\NVar{\rho}}
Property / Symbols used: regular Coulomb radial function / qualifier
 
xml-id: C33.S2.E3.m2abdec
Property / Symbols used
 
Property / Symbols used: Q12668 / rank
 
Normal rank
Property / Symbols used: Q12668 / qualifier
 
Defining formula:

η 𝜂 {\displaystyle{\displaystyle\eta}}

\eta
Property / Symbols used: Q12668 / qualifier
 
xml-id: C33.S1.XMD7.m1cdec
Property / Symbols used
 
Property / Symbols used: Q12691 / rank
 
Normal rank
Property / Symbols used: Q12691 / qualifier
 
Defining formula:

ω 𝜔 {\displaystyle{\displaystyle\omega}}

\omega
Property / Symbols used: Q12691 / qualifier
 
xml-id: C33.S12.XMD5.m1dec

Latest revision as of 14:30, 2 January 2020

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DLMF:33.12.E6
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    Statements

    Defining formula
    F 0 ⁡ ( η , 2 ⁢ η ) 3 - 1 / 2 ⁢ G 0 ⁡ ( η , 2 ⁢ η ) ∼ Γ ⁡ ( 1 3 ) ⁢ ω 1 / 2 2 ⁢ π ⁢ ( 1 ∓ 2 35 ⁢ Γ ⁡ ( 2 3 ) Γ ⁡ ( 1 3 ) ⁢ 1 ω 4 - 8 2025 ⁢ 1 ω 6 ∓ 5792 46 06875 ⁢ Γ ⁡ ( 2 3 ) Γ ⁡ ( 1 3 ) ⁢ 1 ω 10 - ⋯ ) , asymptotic-expansion FRACOP regular-Coulomb-F 0 𝜂 2 𝜂 superscript 3 1 2 irregular-Coulomb-G 0 𝜂 2 𝜂 Euler-Gamma 1 3 superscript 𝜔 1 2 2 𝜋 minus-or-plus minus-or-plus 1 2 35 Euler-Gamma 2 3 Euler-Gamma 1 3 1 superscript 𝜔 4 8 2025 1 superscript 𝜔 6 5792 46 06875 Euler-Gamma 2 3 Euler-Gamma 1 3 1 superscript 𝜔 10 ⋯ {\displaystyle{\displaystyle{F_{0}\left(\eta,2\eta\right)\atop{3^{-\ifrac{1}{2% }}G_{0}\left(\eta,2\eta\right)}}\sim\frac{\Gamma\left(\frac{1}{3}\right)\omega% ^{1/2}}{2\sqrt{\pi}}\left(1\mp\frac{2}{35}\frac{\Gamma\left(\frac{2}{3}\right)% }{\Gamma\left(\frac{1}{3}\right)}\frac{1}{\omega^{4}}-\frac{8}{2025}\frac{1}{% \omega^{6}}\mp\frac{5792}{46\,06875}\frac{\Gamma\left(\frac{2}{3}\right)}{% \Gamma\left(\frac{1}{3}\right)}\frac{1}{\omega^{10}}-\cdots\right),}}
    0 references
    DLMF id
    DLMF:33.12.E6
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11abdec
    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
    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.m1abdec
    0 references
    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.m2abdec
    0 references
    Q12668
    Defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C33.S1.XMD7.m1cdec
    0 references
    Q12691
    Defining formula
    ω 𝜔 {\displaystyle{\displaystyle\omega}}
    xml-id
    C33.S12.XMD5.m1dec
    0 references
     
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