DLMF:16.15.E3 (Q5265): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

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

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1abdec

Latest revision as of 13:41, 2 January 2020

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DLMF:16.15.E3
No description defined

    Statements

    test
    F 3 ⁡ ( α , α ′ ; β , β ′ ; γ ; x , y ) = Γ ⁡ ( γ ) Γ ⁡ ( β ) ⁢ Γ ⁡ ( β ′ ) ⁢ Γ ⁡ ( γ - β - β ′ ) ⁢ ∬ Δ u β - 1 ⁢ v β ′ - 1 ⁢ ( 1 - u - v ) γ - β - β ′ - 1 ( 1 - u ⁢ x ) α ⁢ ( 1 - v ⁢ y ) α ′ ⁢ d u ⁢ d v , Appell-F-3 𝛼 superscript 𝛼 ′ 𝛽 superscript 𝛽 ′ 𝛾 𝑥 𝑦 Euler-Gamma 𝛾 Euler-Gamma 𝛽 Euler-Gamma superscript 𝛽 ′ Euler-Gamma 𝛾 𝛽 superscript 𝛽 ′ subscript double-integral Δ superscript 𝑢 𝛽 1 superscript 𝑣 superscript 𝛽 ′ 1 superscript 1 𝑢 𝑣 𝛾 𝛽 superscript 𝛽 ′ 1 superscript 1 𝑢 𝑥 𝛼 superscript 1 𝑣 𝑦 superscript 𝛼 ′ 𝑢 𝑣 {\displaystyle{\displaystyle{F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{% \prime};\gamma;x,y\right)=\frac{\Gamma\left(\gamma\right)}{\Gamma\left(\beta% \right)\Gamma\left(\beta^{\prime}\right)\Gamma\left(\gamma-\beta-\beta^{\prime% }\right)}\iint_{\Delta}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-v)^{\gamma-% \beta-\beta^{\prime}-1}}{(1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}\mathrm{d}u% \mathrm{d}v,}}
    0 references
    DLMF id
    DLMF:16.15.E3
    0 references
    constraint
    ℜ ⁡ ( γ - β - β ′ ) > 0 𝛾 𝛽 superscript 𝛽 ′ 0 {\displaystyle{\displaystyle\Re(\gamma-\beta-\beta^{\prime})>0}}
    0 references
    ℜ ⁡ β > 0 𝛽 0 {\displaystyle{\displaystyle\Re\beta>0}}
    0 references
    ℜ ⁡ β ′ > 0 superscript 𝛽 ′ 0 {\displaystyle{\displaystyle\Re\beta^{\prime}>0}}
    0 references
    Symbols used
    third Appell function
    test
    Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "XMLReader::XML(): Argument #1 ($source) cannot be empty"): {\displaystyle \AppellF{3}@{\NVar{\alpha}}{\NVar{\alpha^{\prime}}}{\NVar{\beta}}{\NVar{\beta^% {\prime}}}{\NVar{\gamma}}{\NVar{x}}{\NVar{y}}}
    xml-id
    C16.S13.E3.m2adec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2abdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
    0 references
     
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