DLMF:33.12.E7 (Q9605): Difference between revisions

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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.m1adec

Latest revision as of 14:30, 2 January 2020

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DLMF:33.12.E7
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    Statements

    Defining formula
    F 0 ′ ⁡ ( η , 2 ⁢ η ) 3 - 1 / 2 ⁢ G 0 ′ ⁡ ( η , 2 ⁢ η ) ∼ Γ ⁡ ( 2 3 ) 2 ⁢ π ⁢ ω 1 / 2 ⁢ ( ± 1 + 1 15 ⁢ Γ ⁡ ( 1 3 ) Γ ⁡ ( 2 3 ) ⁢ 1 ω 2 ± 2 14175 ⁢ 1 ω 6 + 1436 23 38875 ⁢ Γ ⁡ ( 1 3 ) Γ ⁡ ( 2 3 ) ⁢ 1 ω 8 ± ⋯ ) , asymptotic-expansion FRACOP diffop regular-Coulomb-F 0 1 𝜂 2 𝜂 superscript 3 1 2 diffop irregular-Coulomb-G 0 1 𝜂 2 𝜂 Euler-Gamma 2 3 2 𝜋 superscript 𝜔 1 2 plus-or-minus plus-or-minus plus-or-minus 1 1 15 Euler-Gamma 1 3 Euler-Gamma 2 3 1 superscript 𝜔 2 2 14175 1 superscript 𝜔 6 1436 23 38875 Euler-Gamma 1 3 Euler-Gamma 2 3 1 superscript 𝜔 8 ⋯ {\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{2}{3}\right)}{2% \sqrt{\pi}\omega^{1/2}}\left(\pm 1+\frac{1}{15}\frac{\Gamma\left(\frac{1}{3}% \right)}{\Gamma\left(\frac{2}{3}\right)}\frac{1}{\omega^{2}}\pm\frac{2}{14175}% \frac{1}{\omega^{6}}+\frac{1436}{23\,38875}\frac{\Gamma\left(\frac{1}{3}\right% )}{\Gamma\left(\frac{2}{3}\right)}\frac{1}{\omega^{8}}\pm\cdots\right),}}
    0 references
    DLMF id
    DLMF:33.12.E7
    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.m2aadec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    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.m1acdec
    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.m2acdec
    0 references
    Q12668
    Defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C33.S1.XMD7.m1ddec
    0 references
    Q12691
    Defining formula
    ω 𝜔 {\displaystyle{\displaystyle\omega}}
    xml-id
    C33.S12.XMD5.m1adec
    0 references
     
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