DLMF:33.7.E1 (Q9538): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1adec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2adec
Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: regular Coulomb radial function / rank
 
Normal rank
Property / Symbols used: regular Coulomb radial function / qualifier
 
test:

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.m2adec
Property / Symbols used
 
Property / Symbols used: Q12666 / rank
 
Normal rank
Property / Symbols used: Q12666 / qualifier
 
test:

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

\ell
Property / Symbols used: Q12666 / qualifier
 
xml-id: C33.S1.XMD2.m1dec
Property / Symbols used
 
Property / Symbols used: Q12667 / rank
 
Normal rank
Property / Symbols used: Q12667 / qualifier
 
test:

ρ 𝜌 {\displaystyle{\displaystyle\rho}}

\rho
Property / Symbols used: Q12667 / qualifier
 
xml-id: C33.S1.XMD5.m1dec
Property / Symbols used
 
Property / Symbols used: Q12668 / rank
 
Normal rank
Property / Symbols used: Q12668 / qualifier
 
test:

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

\eta
Property / Symbols used: Q12668 / qualifier
 
xml-id: C33.S1.XMD7.m1dec

Latest revision as of 13:24, 2 January 2020

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DLMF:33.7.E1
No description defined

    Statements

    test
    F ℓ ⁡ ( η , ρ ) = ρ ℓ + 1 ⁢ 2 ℓ ⁢ e i ⁢ ρ - ( π ⁢ η / 2 ) | Γ ⁡ ( ℓ + 1 + i ⁢ η ) | ⁢ ∫ 0 1 e - 2 ⁢ i ⁢ ρ ⁢ t ⁢ t ℓ + i ⁢ η ⁢ ( 1 - t ) ℓ - i ⁢ η ⁢ d t , regular-Coulomb-F ℓ 𝜂 𝜌 superscript 𝜌 ℓ 1 superscript 2 ℓ superscript 𝑒 imaginary-unit 𝜌 𝜋 𝜂 2 Euler-Gamma ℓ 1 imaginary-unit 𝜂 superscript subscript 0 1 superscript 𝑒 2 imaginary-unit 𝜌 𝑡 superscript 𝑡 ℓ imaginary-unit 𝜂 superscript 1 𝑡 ℓ imaginary-unit 𝜂 𝑡 {\displaystyle{\displaystyle F_{\ell}\left(\eta,\rho\right)=\frac{\rho^{\ell+1% }2^{\ell}e^{\mathrm{i}\rho-(\pi\eta/2)}}{|\Gamma\left(\ell+1+\mathrm{i}\eta% \right)|}\int_{0}^{1}e^{-2\mathrm{i}\rho t}t^{\ell+\mathrm{i}\eta}(1-t)^{\ell-% \mathrm{i}\eta}\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:33.7.E1
    0 references
    Symbols used
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    regular Coulomb radial function
    test
    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.m2adec
    0 references
    Q12666
    test
    ℓ ℓ {\displaystyle{\displaystyle\ell}}
    xml-id
    C33.S1.XMD2.m1dec
    0 references
    Q12667
    test
    ρ 𝜌 {\displaystyle{\displaystyle\rho}}
    xml-id
    C33.S1.XMD5.m1dec
    0 references
    Q12668
    test
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C33.S1.XMD7.m1dec
    0 references
     
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