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DLMF:16.15.E2 - MaRDI portal

DLMF:16.15.E2 (Q5961)

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DLMF:16.15.E2
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    Statements

    DLMF defining formula
    F 2 ⁡ ( α ; β , β ′ ; γ , γ ′ ; x , y ) = Γ ⁡ ( γ ) ⁢ Γ ⁡ ( γ ′ ) Γ ⁡ ( β ) ⁢ Γ ⁡ ( β ′ ) ⁢ Γ ⁡ ( γ - β ) ⁢ Γ ⁡ ( γ ′ - β ′ ) ⁢ ∫ 0 1 ∫ 0 1 u β - 1 ⁢ v β ′ - 1 ⁢ ( 1 - u ) γ - β - 1 ⁢ ( 1 - v ) γ ′ - β ′ - 1 ( 1 - u ⁢ x - v ⁢ y ) α ⁢ d u ⁢ d v , Appell-F-2 𝛼 𝛽 superscript 𝛽 ′ 𝛾 superscript 𝛾 ′ 𝑥 𝑦 Euler-Gamma 𝛾 Euler-Gamma superscript 𝛾 ′ Euler-Gamma 𝛽 Euler-Gamma superscript 𝛽 ′ Euler-Gamma 𝛾 𝛽 Euler-Gamma superscript 𝛾 ′ superscript 𝛽 ′ superscript subscript 0 1 superscript subscript 0 1 superscript 𝑢 𝛽 1 superscript 𝑣 superscript 𝛽 ′ 1 superscript 1 𝑢 𝛾 𝛽 1 superscript 1 𝑣 superscript 𝛾 ′ superscript 𝛽 ′ 1 superscript 1 𝑢 𝑥 𝑣 𝑦 𝛼 𝑢 𝑣 {\displaystyle{\displaystyle{F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,% \gamma^{\prime};x,y\right)=\frac{\Gamma\left(\gamma\right)\Gamma\left(\gamma^{% \prime}\right)}{\Gamma\left(\beta\right)\Gamma\left(\beta^{\prime}\right)% \Gamma\left(\gamma-\beta\right)\Gamma\left(\gamma^{\prime}-\beta^{\prime}% \right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-% u)^{\gamma-\beta-1}(1-v)^{\gamma^{\prime}-\beta^{\prime}-1}}{(1-ux-vy)^{\alpha% }}\mathrm{d}u\mathrm{d}v,}}
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    constraint
    ℜ ⁡ γ > ℜ ⁡ β > 0 𝛾 𝛽 0 {\displaystyle{\displaystyle\Re\gamma>\Re\beta>0}}
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    ℜ ⁡ γ ′ > ℜ ⁡ β ′ > 0 superscript 𝛾 ′ superscript 𝛽 ′ 0 {\displaystyle{\displaystyle\Re\gamma^{\prime}>\Re\beta^{\prime}>0}}
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    Symbols used
    second Appell function
    DLMF defining formula
    F 2 ⁡ ( α ; β , β ′ ; γ , missing ; missing , missing ) Appell-F-2 𝛼 𝛽 superscript 𝛽 ′ 𝛾 missing missing missing {\displaystyle{\displaystyle{F_{2}}\left(\NVar{\alpha};\NVar{\beta},\NVar{% \beta^{\prime}};\NVar{\gamma},missing;missing,missing\right)\)\@add@PDF@RDFa@triples\end{document}}}
    xml-id
    C16.S13.E2.m2adec
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    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aadec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aadec
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    real part
    DLMF defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references

    Identifiers

     
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