DLMF:16.15.E1 (Q5263): Difference between revisions

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Property / constraint
 

ℜ ⁡ ( γ - α ) > 0 𝛾 𝛼 0 {\displaystyle{\displaystyle\Re(\gamma-\alpha)>0}}

\realpart@@{(\gamma-\alpha)}>0
Property / constraint: ℜ ⁡ ( γ - α ) > 0 𝛾 𝛼 0 {\displaystyle{\displaystyle\Re(\gamma-\alpha)>0}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: first Appell function / rank
 
Normal rank
Property / Symbols used: first Appell function / qualifier
 
Defining formula:

F 1 ⁡ ( α ; β , β ′ ; γ ; missing , missing ) Appell-F-1 𝛼 𝛽 superscript 𝛽 ′ 𝛾 missing missing {\displaystyle{\displaystyle{F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{% \beta^{\prime}};\NVar{\gamma};missing,missing\right)\)\@add@PDF@RDFa@triples\end{document}}}

\AppellF{1}@{\NVar{\alpha}}{\NVar{\beta}}{\NVar{\beta^{\prime}}}{\NVar{\gamma}% }{\NVar{x}}{\NVar{y}}
Property / Symbols used: first Appell function / qualifier
 
xml-id: C16.S13.E1.m2adec
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
Defining formula:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

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: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1adec

Latest revision as of 14:41, 2 January 2020

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English
DLMF:16.15.E1
No description defined

    Statements

    Defining formula
    F 1 ⁡ ( α ; β , β ′ ; γ ; x , y ) = Γ ⁡ ( γ ) Γ ⁡ ( α ) ⁢ Γ ⁡ ( γ - α ) ⁢ ∫ 0 1 u α - 1 ⁢ ( 1 - u ) γ - α - 1 ( 1 - u ⁢ x ) β ⁢ ( 1 - u ⁢ y ) β ′ ⁢ d u , Appell-F-1 𝛼 𝛽 superscript 𝛽 ′ 𝛾 𝑥 𝑦 Euler-Gamma 𝛾 Euler-Gamma 𝛼 Euler-Gamma 𝛾 𝛼 superscript subscript 0 1 superscript 𝑢 𝛼 1 superscript 1 𝑢 𝛾 𝛼 1 superscript 1 𝑢 𝑥 𝛽 superscript 1 𝑢 𝑦 superscript 𝛽 ′ 𝑢 {\displaystyle{\displaystyle{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,% y\right)=\frac{\Gamma\left(\gamma\right)}{\Gamma\left(\alpha\right)\Gamma\left% (\gamma-\alpha\right)}\int_{0}^{1}\frac{u^{\alpha-1}(1-u)^{\gamma-\alpha-1}}{(% 1-ux)^{\beta}(1-uy)^{\beta^{\prime}}}\mathrm{d}u,}}
    0 references
    DLMF id
    DLMF:16.15.E1
    0 references
    constraint
    ℜ ⁡ α > 0 𝛼 0 {\displaystyle{\displaystyle\Re\alpha>0}}
    0 references
    ℜ ⁡ ( γ - α ) > 0 𝛾 𝛼 0 {\displaystyle{\displaystyle\Re(\gamma-\alpha)>0}}
    0 references
    Symbols used
    first Appell function
    Defining formula
    F 1 ⁡ ( α ; β , β ′ ; γ ; missing , missing ) Appell-F-1 𝛼 𝛽 superscript 𝛽 ′ 𝛾 missing missing {\displaystyle{\displaystyle{F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{% \beta^{\prime}};\NVar{\gamma};missing,missing\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C16.S13.E1.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
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
     
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